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Algorithms for Nonlinear Finite Element-based

Modeling of Soft-tissue Deformation and

Cutting

ALGORITHMS FOR NONLINEAR FINITE ELEMENT-BASED

MODELING OF SOFT-TISSUE DEFORMATION AND CUTTING

BY

BASSMA GHALI, B.Eng.

A THESIS

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING

AND THE SCHOOL OF GRADUATE STUDIES

OF MCMASTER UNIVERSITY

IN PARTIAL FULFILMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

c© Copyright by Bassma Ghali, July 2008

All Rights Reserved

Master of Applied Science (2008) McMaster University

(Electrical & Computer Engineering) Hamilton, Ontario, Canada

TITLE: Algorithms for Nonlinear Finite Element-based Mod-

eling of Soft-tissue Deformation and Cutting

AUTHOR: Bassma Ghali

B.Eng., (Computer Engineering)

Ryerson University, Toronto, Ontario, Canada

SUPERVISOR: Dr. Shahin Sirouspour

NUMBER OF PAGES: xiii, 133

ii

To my family

Abstract

Advances in robotics and information technology are leading to the development

of virtual reality-based surgical simulators as an alternative to the conventional

means of medical training. Modeling and simulation of medical procedures also

have numerous applications in pre-operative and intra-operative surgical plan-

ning as well as robotic (semi)-autonomous execution of surgical tasks.

Surgical simulation requires modeling of human soft-tissue organs. Soft-tissues

exhibit geometrical and material nonlinearities that should be taken into account

for realistic modeling of the deformations and interaction forces between the sur-

gical tool and tissues during medical procedures. However, most existing work

in the literature, particularly for modeling of cutting, use linear deformation mod-

els. In this thesis, modeling of two common surgical tasks, i.e. palpation and

cutting, using nonlinear modeling techniques has been studied. The complicated

mechanical behavior of soft-tissue deformation is modeled by considering both

geometrical and material nonlinearities. Large deformations are modeled by em-

ploying a nonlinear strain measure, the Green-Lagrange strain tensor, and a non-

linear stress-strain curve is employed by using an Ogden-based hyperelastic con-

stitutive equation. The incompressible property of soft-tissue material during the

deformation is enforced by modifying the strain energy function to include a term

iv

that penalizes changes in the object’s area/volume. The problem of simulating the

tool-tissue interactions using nonlinear dynamic analysis is formulated within a

total Lagrangian framework. The finite element method is utilized to discretize

the deformable object model in space and an explicit time integration is employed

to solve for the resulting deformations.

In this thesis, the nonlinear finite element analysis with the Ogden-based con-

stitutive equation has also been applied to the modeling of soft-tissue cutting. El-

ement separation and node snapping are used to create a cut in the mesh that is

close to the tool trajectory. The external force applied on the object along the tool

direction is used as a physical cutting criterion. The possibility of producing de-

generated elements by node snapping that can cause numerical instability in the

simulation is eliminated by remeshing the local elements when badly shaped ele-

ments are generated. The remeshing process involves retriangulation of the local

elements using the Delaunay function and/or moving a node depending on what

is needed in order to generate elements with the required quality.

Extensive simulations have been carried out in order to evaluate and demon-

strate the effectiveness of the proposed modeling techniques and the results are

reported in the thesis. A two-dimensional object with a concentrated external force

has been considered in the simulations.

v

Acknowledgements

I, Bassma Ghali, would like to express my gratitude to everyone who contributed

to making my master studies a great experience. I wish to single out my supervisor,

Dr. Shahin Sirouspour, for his great guidance, support and encouragement during

my master studies.

Sincere gratitude to my fellow colleagues in the Haptics, Telerobotics and Com-

putational Vision Lab, Saba Moghimi, Amin Abdossalami, Ramin Mafi, Ivy Zhong,

Pawel Malysz, Ali Shahdi, Sina Niakosari, Brian Moody, Behzad Mahdavikhah,

Mike Kinsner, Peter Kuchnio, Mahyar Fotoohi and Insu Park for being great friends

who made the lab a fun and enjoyable place to study as well as for their help dur-

ing my graduate program.

I would also like to extent special thanks and appreciations to my family and

friends for their continuous support, encouragement and love throughout my live

that helped me excel in my academic studies.

vi

Contents

Abstract iv

Acknowledgements vi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem Statement and Thesis Contributions . . . . . . . . . . . . . . 7

1.3 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Literature Review 11

2.1 Surgical Simulation Systems . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Modeling of Soft-tissue Deformation . . . . . . . . . . . . . . . . . . . 15

2.2.1 Mathematical Modeling and Constitutive Equations . . . . . . 15

2.2.2 Numerical Modeling and Spacial Discretization Methods . . . 21

2.2.3 Numerical Solution and Time Discretization Methods . . . . . 25

2.3 Modeling Cutting of Soft-tissue . . . . . . . . . . . . . . . . . . . . . . 29

3 Linear Finite Element Modeling 34

3.1 Formulation of the FEM for Linear Analysis . . . . . . . . . . . . . . . 36

vii

3.1.1 The principal of Virtual Work . . . . . . . . . . . . . . . . . . . 36

3.1.2 Derivation of FE Equations for Static System . . . . . . . . . . 38

3.1.3 Derivation of FE Equations for Dynamic System . . . . . . . . 41

3.2 Explicit Numerical Integration Method for Solving Dynamic Equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Nonlinear Finite Element Modeling 47

4.1 Total Lagrangian Formulation with Explicit Time Integration . . . . . 49

4.2 The Deformation Gradient and Large Deformation Stress and Strain

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 Green-Lagrange Strain Tensor . . . . . . . . . . . . . . . . . . 54

4.2.3 Second Piola-Kirchhoff Stress Tensor . . . . . . . . . . . . . . . 56

4.3 Ogden-based Hyperelastic Constitutive Model . . . . . . . . . . . . . 57

5 Modeling of Soft-tissue Cutting 62

5.1 Cutting Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Elements Separation and Node Snapping . . . . . . . . . . . . . . . . 66

5.2.1 Determination of Separated Elements and Snapped Node . . 66

5.2.2 Local Remeshing . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Mesh Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 Force Feedback Calculations . . . . . . . . . . . . . . . . . . . . . . . . 76

6 Deformation and Cutting Simulation: Algorithms and Results 79

6.1 Steps and Results of 2D Explicit Dynamic Linear Finite Element Al-

gorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

viii

6.1.1 Off-line Precomputations and Initializations . . . . . . . . . . 81

6.1.2 Time Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Steps and Results of 2D Explicit Dynamic Nonlinear Finite Element

Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.1 Elemental Nodal Elastic Force Vectors Calculation . . . . . . . 93

6.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3 Steps and Results of 2D Soft-tissue Cutting Simulations . . . . . . . . 103

6.3.1 Main Blocks in the Soft-tissue Cutting Algorithm . . . . . . . 103

6.3.2 Simulation Results of Soft-tissue Cutting . . . . . . . . . . . . 106

7 Conclusions and Future Work 119

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

ix

List of Figures

1.1 LapVRTM virtual reality surgical simulator. ”Reproduces by per-

mission of Immersion Corporation. Copyright c©2008 Immersion

Corporation. All rights reserved.” . . . . . . . . . . . . . . . . . . . . 3

1.2 The da Vincir Robotic Surgery System. c©[2008] Intuitive Surgical,

Inc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Block diagram of a general surgical simulator. . . . . . . . . . . . . . 5

2.1 Examples of haptic devices a) PHANTOMr Premium by SensAble

Technologies, Inc. b) 3-DOF Planer Pantograoh System by Quanser

Consulting Inc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 A general two-dimensional body and the finite element mesh gen-

erated after discretization . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Time discretization of the deformation field in the central difference

method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1 Flow chart of the algorithm employed to perform cutting . . . . . . . 63

5.2 Illustration of determination of separated elements and node snap-

ping: (A) Before node snapping. (B) After node snapping . . . . . . . 67

5.3 Flow chart of the local remeshing algorithm . . . . . . . . . . . . . . . 69

x

5.4 Examples of degenerated triangular elements: (A) Small edge length.

(B) Small area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Example to illustrate local remeshing with retriangulation: (A) Be-

fore node snapping. (B) Degenerated elements creation after node

snapping. (C) Retriangulation of local elements. . . . . . . . . . . . . 71

5.6 Example to illustrate local remeshing with snapped node reposition-

ing: (A) Before node snapping. (B) Degenerated elements creation

after node snapping. (C,D,E,F) Local elements remeshing by retri-

angulation and snapped node repositioning. . . . . . . . . . . . . . . 72

5.7 Illustration of local remeshing by snapped node repositioning with

orthogonal projection: (A) Before node snapping. (B) Degenerated

elements creation after node snapping. (C) Local elements remesh-

ing by snapped node repositioning (D) Local elements remeshing

by projecting the new snapped node on the cutting trajectory. . . . . 74

5.8 Experimental data from liver cutting. Courtesy of [1] . . . . . . . . . 78

6.1 Steps of 2D linear dynamic finite element algorithm . . . . . . . . . . 80

6.2 Meshed circle with the designated node numbers . . . . . . . . . . . 82

6.3 Graphical user interface to specify loading and boundary condition

nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4 Simulation of soft-tissue deformation using linear analysis: (A) Orig-

inal mesh. (B) Deformed mesh after 200 time steps. (C) Deformed

mesh after 400 time steps. (D) Comparison between A and C. . . . . 91

6.5 Displacement vs. external force profile at Node 41 in the Y axis di-

rection for linear FE analysis . . . . . . . . . . . . . . . . . . . . . . . 92

xi

6.6 Steps to calculate nodal elastic force for each element . . . . . . . . . 94

6.7 Simulation of soft-tissue deformation using nonlinear analysis: (A)

Original mesh. (B) Deformed mesh after 200 time steps. (C) De-

formed mesh after 400 time steps. (D) Comparison between A and

C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.8 Simulation of soft-tissue relaxation from a deformed configuration

using nonlinear analysis: (A) Deformed mesh. (B) Mesh configura-

tion 50 time steps after removing the load. (C) Mesh relaxed to the

original configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.9 Comparison between the deformations obtained using linear and

nonlinear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.10 Displacement vs. external force profile on Node 41 in the Y axis

direction for nonlinear FE analysis . . . . . . . . . . . . . . . . . . . . 102

6.11 Main blocks of the cutting algorithm . . . . . . . . . . . . . . . . . . . 104

6.12 Case 1 simulation of soft-tissue cutting using nonlinear analysis: (A)

Original mesh when first contact detected. (B) First contact remesh-

ing. (C) Deformed mesh before the first cut. (D) Node snapping and

elements separation. (E) New loading before second cut. (F) Node

snapping and elements separation. . . . . . . . . . . . . . . . . . . . . 108

6.13 Case1 simulation of soft-tissue cutting after few successive cuts: (A)

Mesh after 350 time steps. (B) A zoomed figure of part A to show

separated elements on the cut path. . . . . . . . . . . . . . . . . . . . . 109

6.14 Case 1 force profile of the external force on the loading node along

the tool direction over 350 time steps . . . . . . . . . . . . . . . . . . . 110

xii




degenerated element generation. (E) Local remeshing. (F) Succes-

sive cuts and mesh configuration after 350 time steps. . . . . . . . . . 112

6.16 A zoomed version of parts D and E from Fig. 6.15 with local ele-

ments illustration: (A) Before remeshing. (B) After remeshing. . . . . 113

6.17 Case 2 force profile of the external force on the loading node in the

tool direction over 350 time steps . . . . . . . . . . . . . . . . . . . . . 114




degenerated element generation. (E) Local remeshing. (F) Succes-

sive cuts and mesh configuration after 350 time steps. . . . . . . . . . 116

6.19 A zoomed version of the local remeshing from part D to E in Fig. 6.18

with local elements illustration: (A) Local elements before remesh-

ing. (B,C,D) The remeshing procedure. (E) Local elements after

remeshing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.20 Case 3 force profile of the external force on the loading node in the

tool direction over 350 time steps . . . . . . . . . . . . . . . . . . . . . 118

xiii

Chapter 1

Introduction

1.1 Motivation

Surgical procedures are complex tasks that require a considerable amount of train-

ing during residency before doctors can start performing actual surgeries. In the

traditional model of training, the resident acquires the skills for performing such

complex tasks through attending as many operations as possible in order to learn

and participate more in the operation each time. Therefore, the resident would

learn by visual cues, repetition and instructive presentations. However, the lim-

ited amount of time available for in-hospital resident training and the high finan-

cial cost of teaching surgical residents in the operating room suggest the need for

an alternative method of training such as virtual surgery training [2, 3] in order

to provide better trained surgeons. Virtual surgery training is particularly needed

for surgical procedures that are more difficult to learn and require extensive prac-

tice. Examples include laparoscopic surgery in which eye-hand coordination is

1

M.A.Sc. Thesis - B. Ghali McMaster - Electrical Engineering

non-intuitive and neurosurgical procedures which involve delicate tissue manip-

ulations that require a great degree of skill and precision to accomplish the task

while avoiding tissue damage.

Recent advances in robotics and information technologies have created new

opportunities in the field of medical training. Virtual Reality (VR)-based simula-

tors are emerging as a promising alternative to the conventional means of train-

ing. They allow surgeons to practice on virtual patients as they would operate on

real patients with realistic sensory feedback. With these simulators, surgical resi-

dents could spend time practicing in a virtual environment repetitively with non-

restricted access to virtual patient and without any risk. The flexibility offered by

the simulator to adjust the tissue properties and operation scenarios without being

concerned about patient safety issues is critical for the training process. They can

also provide training with different complexity levels and define different mea-

sures of performance such as tracking errors and maximum exerted force. Ac-

tive guidance can be provided by these simulators in the form of corrective force-

feedback by the system that would guide the trainee through the task. It can also

be provided in the multi user setting of these simulators through motion super-

vision and corrective forces. In summary, VR training can improve surgical skills

in the operating room which helps surgeons make fewer errors and have shorter

operative times when performing a surgery after enough training [2–4].

In addition to teaching medical students, surgical simulators would allow more

experienced surgeons to keep up with newer surgical techniques and develop-

ments. Laparoscopic surgeries is one of these techniques and it will be seen in the

2


next chapter that many simulators have been developed for training on the essen-

tial skills needed to perform such procedures. LaparoscopyVRTM Virtual Reality

System shown in Fig. 1.1 is an example of these laparoscopic surgical simulators.

Surgeons could also use the simulator as a tool to practice performing surgeries us-

ing surgical robotics [5]. Robotic surgeries require additional and special training

because they are different from the conventional surgery.

Figure 1.1: LapVRTM virtual reality surgical simulator. ”Reproduces by permis-sion of Immersion Corporation. Copyright c©2008 Immersion Corporation. Allrights reserved.”

These VR-based simulators could also be used as an intra-operative and/or

pre-operative planning tool as in [6]. In some procedures, it might be difficult

to predict the consequences of some actions or there could be a number of ac-

tions among which the best possible must be chosen. Therefore, during a surgery,

the surgeon can use the simulator as a strategic planning system to decide the

3


surgical approach that should be taken. Also, pre-operative planning can be per-

formed (e.g. in percutaneous therapy) where the model is used to plan the oper-

ation before the surgery. Patient-specific models can be employed to try different

approaches on the virtual model of the patient and then performing the best ap-

proach during the surgery. Real-time simulation for assisting robots to perform

(semi)-autonomous surgical tasks is another application for surgical simulators.

Robotic surgery is another area that is emerging as an alternative to traditional

surgery. These new technologies provide many advantages for doctors and pa-

tients at the same time [7–9]. Some of these advantages include enhanced opera-

tion precision, increased success rate, reduced trauma to the body, reduced blood

loss, less post-operative pain and less risk of infection [9, 10]. It was also found

that patients who have a robotic surgery tend to have a shorter hospital stay be-

cause of faster recovery which indicates a reduced post-operative health services

and cost. Examples of a robotic surgery system are the da Vincir Surgical System

from Intuitive Surgical Inc. shown in Fig. 1.2 and the Zeusr Surgical System from

Computer Motion Inc. A simulator that can model the tissue-tool interactions dur-

ing the operation can be helpful in evaluating the hardware design in the initial

development of the controllers without any concerns about safety.

A general surgical simulator can consist of a haptic interface to provide force

feedback to the user, a computer system to perform computations and updates

on the virtual object that is being simulated and a monitor to provide graphical

feedback to the user as it is shown in Fig. 1.3. The user interacts with the virtual

model of the organ using the haptic device that functions as surgical tool by spec-

ifying the position of the tool. Depending on the tool position, the virtual model

4


Figure 1.2: The da Vincir Robotic Surgery System. c©[2008] Intuitive Surgical, Inc.

Monitor

Computer

HapticInterface

User

Tool Position

Force/Position

VisualFeedback Graphic Update

Force Feedback

Figure 1.3: Block diagram of a general surgical simulator.

5


gets updated to provide realistic tool-tissue interaction. The graphical update is

presented to the user through the graphical display and the force feedback is pro-

vided through the haptic interface.

Surgical procedures involve interactions with rigid and deformable organs like

bones and soft-tissues, respectively. Modeling tool-soft tissue interactions (defor-

mations and cutting) is generally much more challenging than modeling tool-bone

interactions. For a surgical simulator that involves interactions with a soft organ, a

model of soft-tissues that can provide a realistic graphical and haptic feedback to

the user is needed. The accuracy is also critical in case of pre-operative task plan-

ning and robotic execution of the task. Since soft human ogran’s such as brain,

liver and kidney are found to exhibit a nonlinear mechanical behavior, a model

that takes these complex properties into account should be used. However, most

of the time, many simplifications are assumed and a simple model that does not

provide the realistic behavior needed is used.

In order to find the displacements and forces in the simulated object during

interactions with a surgical tool, the model of the object is discretized in the space

and time domains resulting in a set of algebraic equations in terms of the unknown

displacements and forces. Many techniques are available for discretizing the object

and solving the system of equations. Spatial discretization techniques can provide

different levels of accuracy and speed in the simulation depending on the the-

ory they are based on. Continuum mechanics-based methods are usually utilized

to model tool-tissue interactions since, in general, they provide a better accuracy.

6


Finite Element Method (FEM) is an example of a continuum mechanics-based ap-

proach that is commonly used since it provides results with better fidelity com-

pared to other techniques. Linear models that have been dominantly applied in

the literature to model tool-soft tissue interactions, especially cutting, do not ac-

count for the nonlinear mechanical properties of soft-tissues [11, 12]. Therefore,

nonlinear models should be employed in order to increase the fidelity in simulat-

ing soft-tissue deformations and cutting.

1.2 Problem Statement and Thesis Contributions

This thesis is concerned with the nonlinear modeling of tool-soft tissue interactions

for common but important surgical procedures. The objective of this thesis is not

to build a simulator for a specific task or tissue; instead, the objective is to use non-

linear models that can potentially improve the accuracy in modeling basic surgical

tasks. Since palpation and cutting are among the most common surgical tasks,

modeling of these two interactions is investigated in the thesis. Without loss of

generality, a two-dimensional object that represents a slice of a three-dimensional

soft organ is assumed in the simulations. The modeling involves deforming the

tissue after tool-tissue contact, forming a cut when needed, and calculating a force

feedback sensation. The contributions of the thesis can be summarized as follows:

• Problem 1: Linear elastic FE models, which have been predominantly used

in prior relevant work, are inadequate for realistic modeling of soft-tissue de-

formations and interaction forces with surgical tools. In fact, these models are

only valid for small deformations and are incapable of capturing dominant

7


time-dependent nonlinear behavior of soft-tissues observed in surgery.

Contribution: The Ogden-based nonlinear hyperelastic model presented in [13],

which is found to be the best model to represent the soft-tissue behavior ob-

served in experiments, is used as an energy function of the FE model to cal-

culate soft-tissue deformations. Since soft-tissues are considered rubber like

material with an almost incompressible response, a term is added to the en-

ergy function in order to enforce this condition. The term adds a constrain

on the energy function to keep the area/volume constant during deforma-

tion by using a relatively large bulk modulus as a Lagrange multiplier and a

term that penalizes changes in area/volume. The total Lagrangian formula-

tion is used to formulate the problem and the FEM is employed to spatially

discretize the simulated object. An explicit time integration is utilized to dis-

cretize the dynamic equations of motion in time for the analysis of the system

as in [14] with the addition of damping to the system of equations.

• Problem 2: Modeling soft-tissue cutting, which is a fundamental require-

ment of a surgical simulator, have mostly been done with the assumption

of a simple linear elastic behavior of soft-tissues in the literature in order to

simplify the analysis as it will be shown in Chapter 2.

Contribution: The nonlinear FE model that is mentioned in the contribution

of Problem 1, is employed to model soft-tissue cutting.

• Problem 3: Cutting is a complicated procedure to model as it involves topo-

logical modification in the FE mesh. These changes could cause an increase

in the number of elements or, more critically, numerical instability depending

8


on the technique that is used to implement the cut in the FE mesh. After an

extensive research on existing cutting modeling techniques in the literature,

it is found that node snapping and elements separation in the cutting path

is the best method to model cutting since it does not increase the number of

elements and it creates a cut that is closest to the cut path. However, this

method has the disadvantage of possibly creating degenerated elements that

cause numerical instability of the simulation.

Contribution: An algorithm is developed in order to place the snapped node

in a position that will guarantee stability and generate a cut path that is clos-

est to the tool trajectory.

The proposed techniques for modeling soft-tissue deformation and cutting are

evaluated through simulations that are conducted to demonstrate the performance

of the developed algorithms. Many cases that illustrate important aspects of the

algorithms are shown.

1.3 Organization of the Thesis

The rest of this thesis is organized as follows. Relevant literature pertaining to sur-

gical simulation systems and modeling of soft-tissue deformation and cutting is

presented in Chapter 2. The linear FE modeling process including the formulation

of the FEM for linear analysis and the explicit numerical integration method are

introduced in Chapter 3. Chapters 4 discusses the nonlinear FE modeling which

9


covers the total Lagrangian formulation, the deformation gradient, large deforma-

tion stress and strain measures along with the hyperelastic constitutive model em-

ployed. The algorithm developed for modeling of soft-tissue cutting is presented

in Chapter 5. The results of numerical experiments using the proposed nonlinear

modeling of deformation and cutting are given in Chapter 6. The thesis is con-

cluded in Chapter 7 where some suggestions for future work are also made.

10

Chapter 2

Literature Review

This chapter consists of three main sections. The first section is an overview of

surgical simulation systems. It provides examples of simulators that have been

developed in the past and highlights the importance of haptics in such applica-

tions. The main issues that should be considered when modeling soft-tissue de-

formation are discussed in the second section. These include linear vs. nonlinear

mathematical modeling and constitutive equations; different numerical modeling

and spacial discretization methods; and how to obtain a numerical solution to the

resulting differential equations using time discretization methods. Finally the last

section covers issues in the modeling of soft-tissue cutting and surveys different

methods that have been previously employed to address these issues.

11


2.1 Surgical Simulation Systems

A significant amount of research has been devoted to the simulation of different

surgical procedures. Surgical simulation systems have many applications and pro-

vide surgeons with great advantages as it was explained in Chapter 1. Some of

these applications include training, learning new techniques, planning surgeries,

executing (semi)-autonomous robotic tasks and designing surgical robots. Many

surgical simulators have been developed in the last few years for a wide range

of surgical procedures as in [5, 15–20]. In addition, many companies have man-

ufactured surgical simulators for different procedures such as LaparoscopyVRTM

Surgical Simulation System by Immersion Medical that was shown in Fig. 1.1, LAP

MentorTM by Simbionix, ProMISTM surgical simulator by haptica and TempoSurg

by VOXEL-MAN. One of the main challenges in the development of these sim-

ulators is accurate modeling of tool-tissue interactions which requires the use of

complicated mathematical models. This issue will be considered in depth in the

rest of this thesis. Another important factor in some surgical simulators is provid-

ing the surgeon with a haptic sensation.

Haptics refers to the sense of touch in virtual environments. It allows human

to feel the objects in virtual environments as they are working with real life ob-

jects providing a force/kinesthetic feedback to the user through a motorized joy-

stick [21]. One of the main applications for haptics is the medical field and es-

pecially in surgery simulation and surgical robots. The sense of force/touch can

be very important for the doctor when performing a real surgery using a surgi-

cal robot or training using a surgical simulator. Adding force feedback helps the

12


surgeon or trainer to feel the tool-tissue movement and learn about the mechani-

cal properties of the actual/virtual tissues in contact with the tool [22, 23]. Good

mathematical models and force feedback make the performance of the simulator

closer to reality which is required to have a better training system.

The importance of haptic in surgical simulation can also be noticed by looking

at the literature in the last few years. All the surgical simulation systems men-

tioned above include force feedback and there has been emphasis on the improve-

ment that these simulators accomplish when the sense of touch is added. Also,

most of the work on tissue modeling includes haptics as in [24–27]. The absence of

haptic sensation is considered a disadvantage of surgical robots such as da Vincir

and Zeusr [9]. In these systems, the surgeon performs the surgery on the patient

only depending on vision cues without feeling the forces applied to the tissues or

the texture of the tissues in contact with the surgical tool. This can be very ex-

hausting for the surgeon and can increase the chance of causing errors and tissue

damage during surgery because of excessive forces that the surgeon could apply

without feeling. Therefore, as it is mentioned in [9], many groups are working on

transmitting touch sensation from robotic instruments back to the surgeon.

In the last two-three decades, there has been a lot of research on haptic render-

ing. Many other applications that use the concept of haptic rendering have been

developed over the years. Some of these applications include gaming, pilot train-

ing, teleoperation and robotics. In addition, significant research has been done in

developing haptic devices that give accurate force feedback to the user some of

which are shown in Fig. 2.1.

The direction and magnitude of the force feedback applied by the haptic device

13


A) B)

Figure 2.1: Examples of haptic devices a) PHANTOMr Premium by SensAbleTechnologies, Inc. b) 3-DOF Planer Pantograoh System by Quanser ConsultingInc.

depend on the tool position with respect to the virtual object which is specified by

the user through moving the haptic interface. The force is computed using the

mathematical model employed to represent the virtual object which embodies the

geometrical and mechanical properties of the object in the virtual environment.

Therefore, for a simulator that involves tool interactions with soft-tissue, a model

that represents the mechanical properties of soft-tissues should be used to have a

virtual tissue response that is similar to that of human soft-tissues during surg-

eries. Details regarding different methods to model tool-tissue interactions (de-

formations and forces) and the issues that should be considered to have accurate

deformation modeling are discussed in the next section.

14


2.2 Modeling of Soft-tissue Deformation

Modeling soft-tissue for computer simulation involves many stages [13]. Since the

goal is to simulate soft-tissues during surgeries, it is required to model deforma-

tions and internal/external forces in soft-tissues which are mechanical properties

that can be analyzed using continuum mechanics. The first stage of the modeling

process involves choosing the mathematical model that represents the physical

properties and the constitutive equation that best describes the mechanical prop-

erties of the soft-tissue. The second stage involves choosing a numerical method

to discretize the mathematical model in space in order to be able to solve it nu-

merically using computers. The third stage is solving the discretized equations of

motion to calculate the deformation and interaction forces. The following subsec-

tions present an overview of different methods that have been used in the literature

for each of these stages.

2.2.1 Mathematical Modeling and Constitutive Equations

Linear vs. Nonlinear Modeling of Soft-tissue

Linear analysis of a structural mechanics problem uses an equilibrium equation

based on the principal of virtual work in which the displacement field is a linear

function of the applied load. This analysis includes the assumption of infinitesimal

displacements, i.e., geometrical linearity, and a linear elastic material, i.e., mate-

rial linearity characterized by a linear stress-strain curve [28, 29]. When these as-

sumptions do not apply to the material that is being modeled, a nonlinear analysis

15


should be used. Such analysis could involve material nonlinearity that is charac-

terized by a nonlinear stress-strain relation and/or geometrical nonlinearity which

includes large displacement, large rotations and large strains.

After a comprehensive review of the literature in bio-mechanical modeling of

soft-tissues for surgical simulation, it was found that in many cases these tissues

are modeled as a linear elastic material, e.g., see [6,15,17,18,25,26,30]. Linear elastic

models are popular due to their computational efficiency which allows for real

time simulation. However, linear models are inaccurate in modeling the complex

mechanical behavior of human organs’ soft-tissues because of their material and

geometric nonlinear characteristics. In [11], it is shown that linear elastic models

do not model large deformations correctly as they cause distortion in the object

being modeled; therefore, geometrical nonlinear model was used with a quadratic

strain.

Several methods have been proposed in the literature for simulating complex

physical behavior of soft-tissues. In [31], nonlinear FE was used to simulate a

deformable body. In [32], a deformable model that is characterized with nonlin-

ear elasticity, material large displacement, anisotropy and incompressibility con-

straints was simulated using FEM. In [33], soft-tissues such as kidney were mod-

eled as homogeneous, isotropic, incompressible material with nonlinear elasticity

behavior. Some work has also been done on modeling brain tissue since it is one

of the most complex tissues in the human body. In [12, 34], it was found through

experiments that mechanical behavior of brain tissue is highly nonlinear with a

strong stress-strain rate dependence. Therefore, it was modeled as a single-phase,

nonlinear and viscoelastic material. More recent work in [13, 35–37] show that, in

16


addition to the previously mentioned properties, swine brain tissue is considerably

softer in extension than in compression. This necessitates the use of a more com-

plicated constitutive model in order to take into account the tissue deformations

in tension and in compression as will be discussed later.

Formulations for Nonlinear Analysis

The equilibrium equation of motion for nonlinear analysis can not be solved di-

rectly. Therefore, an approximate solution can be found using a previously calcu-

lated known equilibrium configuration as a reference for variables and linearizing

the resulting equation [29]. When both geometric and material nonlinearities need

to be included, which is the case of soft-tissues of human organs, two types of for-

mulations could be employed depending on which equilibrium configuration is

used as a reference. The first type is the Total Lagrangian (TL) formulation which

is also called Lagrangian formulation. In this approach the original configuration

at time 0 is used as reference for all variables. The second type of formulation is

called the Updated Lagrangian (UL) formulation in which the last calculated con-

figuration is used as a reference for variables. The choice between these two for-

mulations depends mainly on the numerical effectiveness for the modeling process

and on the constitutive law used.

In [14, 31, 38], the TL formulation was used since in this method all variables

are referred to the original configuration allowing for the pre-computation of all

derivatives with respect to the spatial coordinates. This approach results in fewer

mathematical computations compared to the UL formulation in which the deriva-

tives should be recomputed at each time step because the reference configuration

17


changes. Another difference between the two formulations is the simplicity of the

incremental linear strain in UL formulation compared to TL formulation. The ini-

tial displacement effect in the incremental linear strain for TL formulation makes

the strain-displacement matrix more complex [29].

Constitutive Relations

A constitutive equation is a mathematical model that captures the mechanical

properties of the material by describing the relationship between stress and strain.

It enters the equilibrium equations of motion in the calculation for stresses which

in turn are used to compute the internal forces. The TL and UL formulations men-

tioned above take into account nonlinear effects caused by large displacements,

large rotations and large strains. However, accurate modeling of nonlinear behav-

ior of a specific material depends on the choice of the constitutive equation. There

are many constitutive equations available for different types of material in the lit-

erature. The reader is referred to [29, 39] for a comprehensive review of some of

the existing constitutive models.

For elastic materials, which are materials that store energy when loaded, consti-

tutive equations can be used to model linear or nonlinear elasticity depending on

the material behavior. In general, constitutive relations are strain energy functions

that are used to calculate the stresses in a material by determining their derivative

with respect to a strain measure. A constitutive equation consists of constants that

should be found to fit experimental data of the material and is based on a measure

of deformation. The constitutive equation for linear elastic materials is the sim-

plest form of constitutive equations which assumes a linear relationship between

18


the stress and strain. In such a model, which is a generalization of the Hooke’s law,

the stress is a linear function of strain. These equations can not be used to model

the nonlinear stress-strain curve and the finite deformations of soft-tissues [13,29].

Therefore, constitutive relations for nonlinear elasticity are needed for this type of

materials.

Isotropic hyperelastic materials are nonlinear elastic materials that undergo re-

coverable large deformations and for which the stress-strain relationship is de-

rived from a strain energy density function such as rubberlike materials [29,39–41].

Soft-tissues of human organs fall into this category of materials because they have

the mechanical properties of hyperelastic materials. This type of materials includ-

ing soft-tissues are generally almost or completely incompressible which means

that the area/volume of the tissue does not change during deformations. The

incompressibility property is enforced by a constrain on the deformation mea-

sure used which indicates that the ratio of a deformed volume to the original

volume should be equal to one. Also, the strain energy function can be mod-

ified to include this constrain for an almost incompressible material by adding

an extra hydrostatic pressure term to the function which is the basis of a defor-

mation/pressure formulation [29]. These constitutive equations are expressed in

terms of deformation measures such as strain invariants or principal stretch ra-

tios. Many forms of strain energy functions for isotropic hyperelastic materials are

available [29, 39, 41]; however, the most common ones are Ogden [42], Mooney-

Rivlin [43], neo-Hookean [44] and Varga [45] models. The Ogden strain energy

function has the most general form, whereas the other three functions are special

cases of the Ogden function. Other strain energy functions that are not derived

19


from Ogden function are St. Venant-Kirchhoff [32], Yeoh and Arruda-Boyce func-

tions [39, 41].

In [31], Mooney-Rivlin and neo-Hookean were used to model hyperelasticity

in order to handle large deformations and nonlinear elasticity that are common

in biological tissues. In [33], three constitutive models were employed in order

to model the hyperelastic, isotropic and incompressible behavior for kidney and

uterus tissues: neo-Hookean, Mooney-Rivlin, and Fung-Demiray. It was found

that, for kidney tissues, the Fung-Demiray law with an appropriate choice of pa-

rameters produces a better match between numerical and experimental results

when compared with the other two constitutive equations. In [32], St. Venant-

Kirchhoff elasticity was used as an energy function to model nonlinear elasticity

and a constrain was added to the function in order to enforce an anisotropic be-

havior. It was sound that when St. Venant-Kirchhoff model is employed, modeling

the incompressibility property of soft-tissues is difficult and, in most cases, it leads

to instability.

In order to model the nonlinear mechanical behavior of brain tissue, in [12,34],

a strain energy function that is based on Mooney-Rivlin energy function was em-

ployed. The energy was written as a convolution integral with time dependant

coefficients to take into account the time-dependent behavior of the brain tissue.

However in [35], it was found that the brain tissue has different behavior in com-

pression than in tension which can not be properly modeled using the previous

polynomial form of energy function that is based on Mooney-Rivlin. Therefore, an-

other hyperelastic material model based on the Ogden energy function was used

20


since it is a generalization of the Mooney-Rivlin and neo-Hookean energy func-

tions and it allows the use of fractional power of stretches. Also, this model re-

quires fewer constants to be estimated compared to the second order polynomial

hyper-viscoelastic model that was used previously. In [13,35–37], the Ogden-based

hyper-viscoelastic model was utilized to simulate brain tissue deformations. It was

found that this model accurately represents the tissue behavior in tension and com-

pression for a range of strain rate. Because of the complexity of hyper-viscoelastic

model that accounts for most of the computation time, in [13] a simplifying as-

sumption of using a bi-modular Ogden-based hyperelastic equation that is valid

for strain rates commonly observed in surgical procedures was proposed. It was

demonstrated that the stress calculations of the hyperelastic model are in good

agreement with the experimental values for the specified strain rate.

2.2.2 Numerical Modeling and Spacial Discretization Methods

The mathematical model for simulating soft-tissue deformations yields a set of Par-

tial Differential Equations (PDEs) with boundary and initial conditions. In most

cases, it is impossible to find an analytical solution to these equations [46]. There-

fore, in order to be able to solve such a system of differential equations, the math-

ematical model should be discretized in space using a numerical method to find a

numerical solution. In [22, 23, 47–50] different numerical methods for discretizing

deformation models were discussed. These methods could be categorized into two

groups based on the approach followed to deform a surface. The first is geometry-

based modeling techniques in which deformations are solved by geometrically

manipulating the object or the surrounding space. The second is physics-based

21


modeling techniques in which deformations are solved based on the physics in-

volved in the motion and dynamics of tool-tissue interactions. Preference is given

to physics-based techniques over geometry-based techniques when accurate defor-

mation simulation is needed [47,48] because they model the underlying mechanics

of the deformation. In addition, in physics-based modeling techniques, the mag-

nitude and direction of forces applied to each node are automatically computed

which is needed for haptic rendering. Below is an overview of some of the most

commonly used physics-based techniques with their main advantages and disad-

vantages.

Particle-based Method

In this approach, also known as mass-spring method, the object is represented by

a set of point masses (particles) that are connected to each other by a network of

springs and dampers. Each particle has its own position in static systems. In dy-

namic systems, each particle has a velocity and an acceleration component as well.

These points move under the effect of internal and external forces applied on the

object. Many researchers used this technique to model deformations in soft-tissues

as in [19, 51, 52] because of its simplicity and computational efficiency [22]. How-

ever, this method can not be used to model realistic soft-tissue behavior because of

the difficulties associated with integrating tissue properties into the particles and

constructing an optimal mass-spring network [22,30]. In [53], it was stated that due

to the inaccuracy of the mass-spring model, a large number of nodes is required to

generate better results which will cause a large system of matrices in addition to

22


other difficulties that could be encountered such as parameters tuning and model-

ing incompressibility. Another main drawback of mass-spring systems is that the

force computations are strongly related to the topology of the object [23,49]. There-

fore, even thought this method is computationally efficient, other methods such as

FEM are more accurate in modeling deformations of soft-tissues since they rely on

continuum mechanics [23, 52].

Boundary Element Method (BEM)

BME is a numerical computation method that can be used to solve PDEs formu-

lated as integral equations [54]. In this method, only the surface of the object is dis-

cretized and the given boundary conditions are used to fit boundary values into the

integral equations of the boundary nodes in order to solve for the displacements

of the boundary nodes. BEM was used in [17, 53] to model deformable objects

for surgical simulation. It is more computationally efficient than other methods,

including FEM, because the interior of the body is not discretized which reduces

the size of system of equations that should be solved. However, this can be inaccu-

rate when internal deformations and stresses are important and/or when there is a

high surface to volume ratio such as the case with medical applications. Therefore,

when internal properties are important or material with complicated nonlinear be-

havior should be modeled, other volume discretization methods like FEM should

be employed [53]. Also, the fully populated matrices generated in BEM cause the

storage requirement and computational time to grow by the square of the problem

size compared to a linear increase in the FEM using sparse matrices [55].

23


Finite Element Method (FEM)

In FEM, the object is divided into a mesh of elements that are assembled together

through the nodes of these elements in order to find the nodal deformations when

a load is applied using interpolation functions over the mesh [29, 56, 57]. There

are many types of elements and shape functions that can be used and their choice

depends on the problem being solved and the geometry of the object to be dis-

cretized. The properties of each element are formulated into matrices used in the

calculations of the unknown nodes displacements and forces. FEM is one of the

most popular and most effective methods that have been employed to approxi-

mate solutions of PDEs and of integral equations of the continuum mechanics that

govern soft-tissue behavior. The majority of the work conducted on modeling de-

formable objects such as human organs utilized this method, e.g., see [11,13,15,18,

25, 27, 30, 31, 58], even though it is generally more computationally expensive than

the other methods mentioned earlier.

What makes FEM very popular is the continuum-based approach that needs

only few material parameters and the ability to model a multilayered tissues that

exhibit complex nonlinear, viscoelastic and anisotropic behavior [22]. A special

case is the explicit FEM which is also called tensor-mass approach. In this ap-

proach, the physical object is discretized to finite elements just as in the general

FEM, but the masses, damping, and forces are lumped on the nodes of the mesh.

This method is a transition from the mass-spring method to the FEM and is less

computationally expensive than the general FEM. It makes mesh modification in

case of cutting easier to achieve because the calculations are done at the element

level and there is no need to assemble global matrices [14, 49, 59].

24


Meshfree Techniques

Meshfree methods discretize the physical model into a cloud of unconnected points

with masses affecting each other according to some mathematical formulation.

In [48], a survey of different Lagrangian meshfree methods was presented. It is

shown that these methods have been used in many applications including the an-

imation and simulation of deformable bodies. Since these methods avoid using a

mesh, they can provide a solution for the drawbacks of FEM such as remeshing

which is very computationally intensive when needed during simulation. Also

these methods can be utilized for modeling irregular objects where meshing is a

complex task. A meshless numerical technique called the Method of Finite Spheres

(MFS) was used to model physics-based soft-tissue simulation in [60]. It is a gen-

eralization of the FEM that uses the Galerkin formulation to discretize the PDEs

which govern the deformable object. However, the displacement field is approxi-

mated using functions defined over a spherical area around the node [22]. It was

shown that, compared to the solution of FEM, the solution of this method is quite

accurate in the area around the tool tip whereas inaccuracy increases away from

the tool tip. In addition, in [38], the Element Free Galerkin method (EFG) was used

to calculate deformations in soft-tissues.

2.2.3 Numerical Solution and Time Discretization Methods

After Discretizing the system of PDEs of motion for a deformable object using one

of the techniques mentioned above, a system of ordinary differential equations is

obtained. When a static system that does not take inertia and damping forces into

25


consideration is being solved, the system of equations is:

KU = R (2.1)

where K is the stiffness matrix, U is the displacement vector and R is a vector of

external forces. However, when inertia and damping forces are needed to be taken

into account during calculations, which is the case in this thesis, the following

dynamic system of equations should be solved:

MU + DU + KU = R (2.2)

where M and D are mass and damping matrices; U and U are acceleration and

velocity vectors, respectively. The K matrix is constant in the case of a linear elastic

system. However, when a nonlinear elastic material is being analyzed, the K ma-

trix is not constant and it is dependent on the deformation of the object. In order

to solve such a dynamic system of equations, the time-dependent variables have

to be discretized in time. Therefore, these equations have to be solved using a time

integration scheme in which Eq. 2.2 is satisfied at discrete time points separated

by intervals called time steps (4t). The position, velocity and acceleration of each

node in the discretized object are updated at each simulation time step. The se-

lection of the magnitude of this time step is important because a very large value

of time step could cause the result to diverge, while a small value would unnec-

essarily increase the computations. For dynamic systems, these time integration

schemes are divided into two main methods: implicit and explicit. Below is a brief

review of each of these schemes. A detailed explanation of these approaches could

26


be found in [29, 30].

Implicit Integration Methods

In implicit time integration methods, the equilibrium conditions at time t +4t are

used to calculate the deformations at time t +4t. Examples of these methods in-

clude Houbolt, Wilson, and Newmark methods [29]. In [18], the trapezoidal rule

implicit method was employed to solve the linear dynamic equations of model-

ing deformations in brain tissues. The main advantages of these methods include

their unconditional stability and possibility of using large time steps which could

not be used in explicit methods. However, these methods require matrix inver-

sion in order to solve the system of equations and find the deformations of each

node in the mesh. This process is computationally expensive when large system

of equations is needed to model the object. In some cases, the inverse of the matrix

is pre-computed in order to speed up the calculations during time stepping. How-

ever, when nonlinear models are used and/or a topological modification occurs

which require a change in the stiffness matrix, the pre-computation of the inverse

does not help because it has to be recalculated for the new stiffness matrix.

Explicit Integration Methods

In explicit time integration methods, the equilibrium conditions at time t are used

to calculate the deformations at time t + 4t. There are many explicit methods

available and they have been employed extensively for simulating models of dy-

namic deformable objects. In [25], the fourth-order Runge-Kutta was employed

for discretizing the time domain instead of the Euler method since it was found

27


that it can accommodate larger time steps which could lead to a speed-up in the

calculations. In [59], three different explicit methods were examined: first-order

Euler integration, fixed time step fourth-order Runge-Kutta, and three different

formulations of the Verlet algorithms. The Leapfrog Verlet technique was the one

chosen because it was found to give the best results compared to the other two

methods. In [49], the solution of the governing equations was obtained using a

modified-Euler integration method which is an explicit method.

The most common explicit method is the central difference technique which

was employed in [11,14,31,38] for computing soft-tissue deformations. In all these

publications, an explicit method was selected to avoid solving a system of alge-

braic equations at every time step which requires setting up and inverting a large

sparse matrix or using iterative methods. Also, calculations can be done at the

element level when lumped mass and damping matrices are utilized. Therefore,

these methods are more computationally efficient than implicit methods. In ad-

dition, explicit methods and the central difference method in specific make the

treatment of nonlinearities straight forward since they do not require the calcu-

lation of the strain incremental stiffness matrices as it is the case for implicit time

integration methods. Therefore, the central difference method is the most common

explicit method for nonlinear dynamic analysis [14, 29]. A main disadvantage of

explicit integration schemes is that they are only conditionally stable and their sta-

bility depends on the time step chosen for the calculations. The time step has to

be smaller than a critical length specified by the material properties and the size of

elements in the mesh [14, 29, 57].

28


2.3 Modeling Cutting of Soft-tissue

Soft-tissue cutting is one of the common tasks during a surgery. It is also one of the

most complicated procedures to model since it involves a topological modification

of the underlying model. A literature review on techniques for simulating soft-

tissue cutting revealed that most available techniques use linear elastic models to

simplify the analysis for a real-time simulation. For example, linear elastic FE mod-

els were utilized in [25, 59, 61] to simulate cutting in deformable objects. In [62], a

linear FE mesh was employed as a physical model for a deformable object and a

linear mass-spring/particle system was used in the remeshing stage when a topo-

logical modification is needed in order to homogenize the mesh in the vicinity of

the cut and prevent small elements that cause instability. In [63], a linear mass-

spring model was proposed for the modeling of cutting. In [64], a system of point

masses that are connected with linear, semi-linear and nonlinear springs/dampers

was developed for modeling the cutting. The work presented in [49] utilized a

linear elasticity model and a nonlinear strain tensor in order to take into account

large deformations of soft-tissues with a linear stress-strain relation.

These linear elastic models can not generate a realistic haptic feedback and de-

formation of human organs as discussed in the previous section because of the

assumptions of a linear stress-strain relationship and infinitesimal deformations

embedded in the equations of linear elasticity. In [32, 65], nonlinear elasticity of

soft-tissues during deformation and cutting was modeled. However, the energy

functions used, such as St. Venant-Kirchhoff and neo-Hookean energy functions,

were found to be not the best energy functions to represent the nonlinear behavior

in soft-tissues. Therefore, more complex nonlinear formulations that model large

29


deformations should be employed to improve the realism.

Many mesh cutting techniques have been developed in the literature and a sur-

vey of different aspects of these techniques is given in [64]. These techniques in

general try to reduce the number of new elements created after a cut, generate

new elements with good quality, keep a continuous mesh structure between the

elements in the cut path, and create a cut that go through the tool path as close

as possible. The main methods that are used to model cutting include removing

elements [25, 32], subdividing the intersected elements [59, 63, 64, 67, 68], separat-

ing elements along the cut path [49, 61, 62, 65] and refining followed by separating

elements [69].

The element removal technique does not preserve the volume and mass of the

model and requires a very fine mesh to generate a smooth cutting path. The ele-

ment subdivision and refinement followed by separation techniques generate the

best fitting cutting path; however, they increase the number of elements in the

model after each cut which cause a decrease in the performance of the simulation.

The element separation technique does not increase the number of elements in the

model, preserves the mass and area of the model and creates a cut that is the closest

to the path traversed by the tool using node snapping. The only disadvantage of

this method is elements with bad quality (small area or bad shape) could be gen-

erated when selected nodes are snapped to create a cut that fits the tool trajectory.

This could cause a numerical instability in the simulation since the critical time

step decreases and becomes smaller than the utilized time step. The system could

continue to be stable if the time step is changed to a smaller value; however, that

would slow down the computations. Therefore, if a proper method is employed to

30


ensure that degenerated elements are not generated when node snapping is per-

formed, element separation is the best method to model the cutting procedure in

deformable objects.

Simulating cutting in soft-tissues requires a cutting criterion to determine when

the cutting procedure should be performed. This can be a physical criterion, geo-

metrical criterion or both. In [49], geometrical and physical conditions should be

met before a cutting procedure was performed. The geometrical criterion deter-

mines if the user displacement on the surface of the simulated object corresponds

to a cutting attempt or not and the physical criterion checks if the physical interac-

tion between the cutting tool and the object is sufficient to break the object. To this

end, the stress in the object that is subjected to an external load by the cutting tool

was considered the physical criterion and the object was broken when the maxi-

mum stress becomes greater than the material toughness. In [67], if the external

force component in the plane of the tool exceeded a tissue dependent threshold

force, the scalpel would start cutting. The force at the tool tip was used as a cutting

criterion in [66]. When this force reached a threshold value which was obtained

from physical experiments, the cut was created in the mesh. In [61–63], the cut

surface or line was defined only according to the tool movement.

Since the cutting procedure involves a topological modification and node snap-

ping, badly shaped elements with small area can be generated. These elements are

called degenerated elements that can cause numerical instability in the simula-

tion [49, 65]. Therefore, the cutting algorithm should properly handle these cases

to avoid instability. In [49, 61], whenever a degenerated element was detected, it

was removed with some considerations to minimize the effect on the mass of the

31


object. However, not all elements can be removed without larger changes to the

mesh and such approach can violate the principle of mass conservation. In [62,63],

the mesh quality was improved after performing cutting and node snapping by a

sequence of local mesh relaxation steps that moves the interior nodes in order to

have an even spacing between the nodes in the mesh.

Cutting of soft-tissue can be affected by many factors such as the representa-

tion of the tool, the sharpness of the cutting tool, and the speed of applying the

load. In [66], liver cutting experiments demonstrated that the critical force for cut-

ting is dependent on the sharpness of the tool tip and the tool velocity. A round

tip blade was found to have much higher critical force and a longer initial defor-

mation phase before initiating a cut. It was also determined that the cutting force

increases if the tool velocity increases which showed the viscoelastic behavior of

soft-tissues. The cut opening displacement was another issue that was examined

and found to be dependent on the length and depth of the cut as well as the pre-

stresses in the tissue. The relationship between slicing angle, blade edge geometry,

contact length, the fracture force and the applied force were examined through ex-

periments in [70]. In [49], the effect of the tool sharpness was considered by adding

a sharpness factor dependent on the size of the tool edge. The calculated stresses in

the tool-soft tissue contact area are scaled by this factor to generate higher stresses

for sharper tools. Also, the fact that a damage can occur in the cutting area was

added by introducing a damage parameter that increases the effect of loading in

an already cut area.

Meshfree methods which are numerical techniques that do not use a mesh can

32


also be utilized to model cutting instead of mesh-based methods. In [22], a mesh-

free method was proposed and it was claimed that it is a promising technique to

model cutting since it can have the potential of solving some of the problems as-

sociated with remeshing when a mesh-based FEM is employed. In [66], a hybrid

approach to simulate surgical cutting procedures was proposed. The approach

combined a node snapping technique with a physically-based mehsfree computa-

tional scheme.

33

Chapter 3

Linear Finite Element Modeling

In order to simulate a deformable body, a mathematical description of the geome-

try and elasticity is needed to describe the relation between the displacement intro-

duced by the load and the deformations and forces in the body. The displacement

function which assigns a displacement to every point in the deformable body is es-

sentially a mapping with continuous domain and range spaces. The FEM is used in

order to discretize the domain of the displacement function. It is the best method to

describe the physics of deformable objects such as human organs since it is based

on the strong mathematical foundation of continuum mechanics. In the FEM, the

body is divided into a finite number of elements and nodes in order to reduce the

problem to a finite number of unknowns, in this case nodes displacement. This

process is illustrated in Fig. 3.1. After the spatial discretization, the displacement

field within each element is described in terms of nodes displacements using an

interpolation function.

However, the standard FEM can be computationally expensive and not suit-

able for topological modifications because that would require modifying the global

34


-0.1 -0.05 0 0.05 0.1 0.15

-0.1

-0.05

0

0.05

0.1

0.15

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

-0.1 -0.05 0 0.05 0.1 0.15

-0.1

-0.05

0

0.05

0.1

0.15

Bu

SpatialDiscretization

A

X

Y

Bf

Ri

ElementNode

Fixed Nodes

fb

x

fBf

fb

y

i

Figure 3.1: A general two-dimensional body and the finite element mesh generatedafter discretization

matrices of the object. Therefore, an explicit FEM (tensor-mass model) that lumps

masses on the nodes have been used in this thesis since it is more computation-

ally efficient than the original FEM and it can be easily adapted to accommodate

topological changes due to operations such as cutting.

An explicit integration method, the central difference technique, is utilized to

solve the dynamic system of equilibrium equations generated by the finite element

method. Such approach can be computationally more efficient than implicit meth-

ods since it does not need solving a system of equations. The central difference

method is commonly employed for nonlinear dynamic analysis in the literature

since it can easily treat nonlinear deformation models. The main disadvantage of

this method is that it is only conditionally stable which can occasionally impose

a severe restriction on the time step size in the simulation. In particular, a time

step that is smaller than the critical time step should be used to guarantee numer-

ical stability. In the case of topological modification where critical time step could

change due to the changes in the elements, an algorithm should be developed to

ensure that the new elements with the existing simulation time step would not

35


violate the stability condition.

The first section of this chapter provides an overview of the finite element for-

mulation for linear analysis to show how the method is employed to discretize the

equations of motion. It starts by stating the continuous problem that is required

to be solved and the principal of virtual work that is the basis of the FEM. A fo-

cus on the disretization of the continuum mechanics governing equations using

FEM is given next to acquire the discretized equations of motion for a static sys-

tem. Finally, the derivation of the governing equations for a dynamic system is

discussed. In the second section, a description of the central difference method

and its application to solve the governing equations of motion is given. The mate-

rial in this chapter is based on the detailed explanation given in [29] and uses the

same notation.

3.1 Formulation of the FEM for Linear Analysis

3.1.1 The principal of Virtual Work

The derivation of the finite element equilibrium equations for a linear elastic ma-

terial starts with the general elasticity problem that considers the object to be mod-

eled as a two-dimensional linear elastic body A shown in Fig. 3.1. The boundary

of the area A consists of two parts: Bu which is the support of the body and is

fixed with prescribed displacements UBu ; and Bf a part of the boundary that is

subjected to boundary traction fBf . The area A is subjected to body forces fb and

an externally applied load force concentrated on finite number of specific points i

of the area Ri. All these forces have two components corresponding to the X and

36


Y coordinate axes.

The body displacement at any internal point on the object is denoted by U

which also has two components, one for each coordinate axis as follows:

U(X,Y ) = [U V ]T (3.1)

The prescribed displacements at the fixed boundary Bu are given by U = UBu .

The linear strain field corresponding to the displacement field U is:

e = [exx eyy exy]T (3.2)

where exx = ∂U∂X

; eyy = ∂V∂Y

; exy = ∂U∂Y

+ ∂V∂X

The above relation could be rewritten in a matrix form as e = B U where U is

defined in 3.1 and B is the strain-displacement matrix given by:

B =

∂∂X

0

0 ∂∂Y

∂∂Y

∂∂X

(3.3)

For linear elastic materials, the stresses corresponding to the strain tensor given

in Eq. 3.2 are found through Hooke’s law:

τ = Ce = [τxx τyy τxy]T (3.4)

where C is the stress-strain relation matrix.

37


The objective is to find the deformation field U of the body A and the corre-

sponding strains e and stresses τ given the applied loads, boundary conditions

and the material stress-strain law. In order to solve this problem, the principal of

virtual work can be employed to establish the governing differential equations at

the equilibrium, i.e.

∫

A

eT τdA =

∫

A

UT fbdA +

∫

Bf

UBTf fBf dl +

∑i

UiT Ri (3.5)

Eq. 3.5 states that for any body like the one shown in Fig. 3.1 to be at equilibrium,

the total internal virtual work (left hand side of the equation) must be equal to the

total external virtual work (right hand side of the equation) when a small virtual

displacement applied on the body in its state of equilibrium. It should be noted

that the displacement field in Eq. 3.5 is a continuous field and all integrations are

performed over the original area without the effect of imposed virtual displace-

ments.

3.1.2 Derivation of FE Equations for Static System

To derive the discretized equations from the continuous equation given in 3.5,

the finite element discretization scheme is employed. In this approach, the body

shown in Fig 3.1 is divided to discrete triangular finite elements that are intercon-

nected at nodal point. The displacement within each element is assumed to be a

function of the nodal point displacement using an interpolation function. There-

fore, for each element, the displacement field is:

um = Hm u (3.6)

38


where Hm ∈ R2×6 is the displacement interpolation matrix, the superscript m de-

notes element m, and u is a vector of element nodal displacement such that:

u = [u1 v1 u2 v2 u3 v3 . . . un vn]T (3.7)

and n is the number of nodes per element and the numbers 1, 2, . . . n are the lo-

cal numbers of the nodes. Based on Eqs. 3.6 and 3.7, for a three node triangular

element, the displacement and position of any point within the element can be ex-

pressed in terms of the nodal displacements and positions using the interpolation

function as follows:

um =3∑

n=1

hn un ; vm =3∑

n=1

hn vn (3.8)

xm =3∑

n=1

hn xn ; ym =3∑

n=1

hn yn (3.9)

where hn is the interpolation function or the isoparametric coordinate of node n.

The interpolation functions (h1, h2, h3) are the components of the displacement in-

terpolation matrix Hm given in Eq. 3.6 and are functions of nodes coordinates as

will be detailed in Chapter 6.

Given the discretized deformation field in Eq. 3.7, the continuous strain and

stress fields in Eqs. 3.2 and 3.4 can now be written for each element as follows:

em = Bmu (3.10)

τm = Cmem (3.11)

39


whereBm ∈ R3×6 is the strain-displacement matrix of element m and Cm ∈ R3×3 is

the elasticity matrix of element m.

After the discretization, Eq. 3.5 can be rewritten as a sum of integration over

the area of all elements:

k∑m=1

∫

Am

eTmτmdAm =

k∑m=1

∫

Am

uTmfb

mdAm +k∑

m=1

∫

l1m,...,lqm

uBT

m fBmdlm +

∑i

uiT Ri (3.12)

where k is the total number of elements, l1m, . . . , lqm denote the element edges that

are part of the boundary of the object and fBm are boundary traction where the

superscript B includes the whole boundary.

By substituting Eqs. 3.6, 3.10 and 3.11 into Eq. 3.12, the following equation is

obtained:

UT

[k∑

m=1

∫

Am

BTmCmBmdAm

]U =

UT

[k∑

m=1

∫

Am

HTmfb

mdAm +k∑

m=1

∫

l1m,...,lqm

HB Tm fB

mdlm + Rc

](3.13)

where R is a vector of the concentrated load forces applied to the nodes of the mesh

and U is a vector of global nodal displacement of all nodes in the mesh.

To continue, the following notations are defined based on Eq. 3.13:

K =k∑

m=1

Km =k∑

m=1

∫

Am

BTmCmBmdAm (3.14)

where K and Km are the global and element stiffness matrices, respectively and

40


R = Rb + Rl + Rc

=k∑

m=1

Rbm +

k∑m=1

RBm + Rc

=k∑

m=1

∫

Am

HTmfb

mdAm +k∑

m=1

∫

l1m,...,lqm

HB Tm fB

mdlm + Rc

(3.15)

where R is the load vector which consists of body forces Rb, boundary forces RB

and concentrated load forces Rc.

Using Eqs. 3.14 and 3.15, Eq. 3.13 can be rewritten as a system of linear equa-

tions in terms of the unknown nodal displacements as follows:

K U = R (3.16)

3.1.3 Derivation of FE Equations for Dynamic System

Eq. 3.16 represents the static equilibrium equation of the assembled mesh in which

inertia and damping forces are not taken into account. These forces need to be

considered for dynamic simulations in order to model the effect of rapidly applied

loads and the energy dissipation during vibration. Such forces can be incorporated

into the body force vector Rb as follows:

Rb =k∑

m=1

∫

Am

HTm

[fbm − ρmHmU− κmHmU

]dAm (3.17)

where U is a vector of nodes accelerations, ρm is the mass density of element m, U

is a vector of the nodes velocities and κm is the damping property of element m.

41


By defining the mass matrix as:

M =k∑

m=1

Mm =k∑

m=1

∫

Am

ρmHTmHmdAm (3.18)

and the damping matrix as:

D =k∑

m=1

Dm =k∑

m=1

∫

Am

κmHTmHmdAm (3.19)

the equilibrium equation can be rewritten as follows:

MU + DU + KU = R (3.20)

M, D and K ∈ R2N×2N and U, U and U ∈ R2N×1 where N is the number of nodes

in a two dimensional mesh. Eq. 3.20 represents the global governing dynamics of

a linear-elastic finite-element model for soft-tissue deformation constructed based

upon the elemental matrices.These equations must be solved in order to obtain the

nodal deformation vector U as a function of time. The assumption of infinitesimal

displacements has entered the equilibrium equation in the evaluation of the stiff-

ness matrix K and the load vector R since all integrations have been performed

over the original area of the elements and the strain-displacement matrix is as-

sumed to be constant and independent of elements displacements. The other as-

sumption of linear elastic material has entered by the use of a constant stress-strain

matrix C.

42


3.2 Explicit Numerical Integration Method for Solv-

ing Dynamic Equations

A time integration method should be employed to solve the discretized equations

of motion given in Eq. 3.20 and compute the deformation field U. To this end, the

time variable is discretized and the deformation variables are only computed at

sample times separated by4t time steps, as shown in Fig. 3.2. Soft-tissue modeling

requires an efficient numerical scheme when integrating the equations of motion

in the time domain to balance computation speed against numerical accuracy and

simulation stability. Therefore, an explicit time integration method, the central

difference method is used in this thesis.

In the explicit central difference method, the solution from the previous time

step (time=t) is used in order to find the displacement field at the next time step

(time=t+4t). Therefore, the global system of discretized equations of motion that

is used to solve for the displacement at t +4t is:

M tU + D tU + K tU = tR (3.21)

where superscript t means that the solution at time t is considered.

Fig. 3.2 shows the time discretization of the displacement in the central differ-

ence method. Difference formulas for approximating the velocity and acceleration

in terms of displacements can be derived as follows:

V elocity : tU =1

24t(t+4tU−t−4t U) (3.22)

43


Time

Deformation(U)

TT-∆t T+∆t

∆t

2∆t

T+0.5∆tT-0.5∆t

TÙ

T+0.5∆tÙT-0.5∆tÙ

Figure 3.2: Time discretization of the deformation field in the central differencemethod

Acceleration : tU =1

4t(t+ 1

24tU−t− 1

24t U)

=1

4t

(t+4tU−t U

4t−

tU−t−4t U4t

)

=1

4t2(t+4tU− 2tU +t−4t U)

(3.23)

By substituting Eqs. 3.22 and 3.23 into the system of dynamic equations in 3.21,

the following relation is obtained:

M4t2

(t+4tU− 2tU +t−4t U) +D

24t(t+4tU−t−4t U) + K tU = tR (3.24)

In order to solve for the deformation field in the next time step at t+4t, Eq. 3.24

can be rewritten as follows:

(M4t2

+D

24t

)t+4tU =t R−

(K− 2M

4t2

)tU−

(M4t2

− D24t

)t−4tU (3.25)

44


To reduce the calculations at each time step, a lumped (diagonal) mass M and

damping D matrices can be employed as suggested in [25, 30]. This technique

assumes the damping and mass effects are concentrated at the nodes only and the

resulting viscous damping is with respect to ground.

It can be observed from Eq. 3.25 that when an explicit integration method is

used in conjunction with diagonal mass and damping matrices, no matrix inver-

sion is needed and calculations can be performed at the element-level. This can

be clearly seen in Eq. 3.25 since the force calculation on the right hand side can

be done independently for each element, and the left hand side is diagonal. This

significantly reduces the computational cost of each time step compared to that of

implicit methods. Therefore, there is no need to assemble the stiffness matrix and

the computations of the elastic forces are performed for each element separately

and contributions of each element are summed to generate the elastic force vector

as follows:

K tU =∑

e

KetU =

∑e

tFe (3.26)

where e denotes the eth element of the mesh and F is the elastic force. F for each ”el-

ement” (Fe)can be computed based on the element stiffness matrix and the nodal

displacements of the element. After generating the vectors of the right hand side

of Eq. 3.25, the large global Eq. 3.25 can be divided into simple independent equa-

tions, two for each node. Therefore, each node’s displacement can be calculated

separately using the following equation:

t+4tUi =

(Mii

4t2+

Dii

24t

)−1 [tRi − Fi +

(2Mii

4t2

)tUi −

(Mii

4t2− Dii

24t

)t−4tUi

]

(3.27)

45


for i=1, 2,. . . , 2N where i denotes the ith component of the vector and ii denotes the

diagonal entry in the ith row of the matrix.

46

Chapter 4

Nonlinear Finite Element Modeling

The linear finite element formulation explained in the last chapter assumes in-

finitesimal displacements of the object and a linear elastic material. These assump-

tions, however, often do not apply to soft-tissues of human organs as discussed

in Chapter 2. Nonlinear models are more suitable for simulating the deforma-

tion response for such soft-tissues. Nonlinear models can account for material and

kinematic nonlinear effects that often characterize human organ soft-tissues defor-

mations by employing suitable constitutive equations and large deformation stress

and strain measures.

As it was mentioned in Chapter 2, total and updated Lagrangian formulations

are the main formulations for nonlinear analysis [29]. In this thesis, the total La-

grangian formulation is used where stresses and strains are measured with respect

to the original configuration. This choice allows for pre-computation of most spa-

tial derivatives before the commencement of the time-stepping procedure and is

capable of handling both geometric and material nonlinearities. Geometric non-

linearity is included by using a nonlinear strain measure in order to properly

47


model large deformations. The material nonlinearity is considered by using a

nonlinear stress-strain relation based on the Ogden-based hyperelastic constitu-

tive model [13, 42] as an energy function for the system. To achieve material in-

compressibility in the simulations, the energy function is modified by adding a

term that would penalize volumetric (area in our case) variations in the deformed

object.

The dynamic analysis is performed with an explicit time integration method as

opposed to implicit methods. Compared to the formulation for implicit time in-

tegration, the development of the nonlinear formulation for explicit dynamic time

integration is rather straightforward since it requires no iterations and no strain in-

cremental stiffness matrices calculations. The only potential disadvantage of this

approach is the conditional stability of the simulation which imposes a restriction

on the size of the time step employed. The central difference method explained in

Sec 3.2 is the explicit time integration employed.

The first section of this chapter provides an overview of the total Lagrangian

formulation used in the nonlinear analysis to derive the discretized equations of

motion. The second section introduces the stress and strain measures utilized in

this thesis to enable modeling large deformations of the object. Finally, an illus-

tration of the Ogden-based hyperelastic constitutive equation and the implemen-

tation of the incompressible condition is given in the last section. The material and

notations used in this chapter are mainly based on [29].

48


4.1 Total Lagrangian Formulation with Explicit Time

Integration

The FEM is a continuum-based approach in which the governing continuum me-

chanics equations are developed based on the principal of virtual work as dis-

cussed in Sec. 3.1.1. In this chapter, the same principal is employed with the addi-

tion of nonlinear effects including large displacement, rotations and strains as well

as a nonlinear stress-strain relationship. Therefore, the principal of virtual work

given in Eq. 3.5 and the analysis demonstrated in the previous chapter are the ba-

sis of the nonlinear analysis given below. Also, Fig. 3.1 can be employed in this

chapter for the definition of the body under consideration since a similar notation

is used.

In order to find the displacement field of a body corresponding to an applied

load in nonlinear analysis, an incremental formulation is employed and time vari-

ables are used to describe the loading and motion of the body. The goal is to com-

pute the displacement field of the body at discrete time points separated by an

increment 4t. To obtain the solution at time t +4t, the solution for all time steps

from time 0 to time t are assumed to be known. This analysis is considered a La-

grangian formulation since all particles of the body are followed in their motion

from the original to the current configuration.

The principal of virtual work for the body at time t in an incremental La-

grangian formulation should be considered since the dynamic analysis is performed

with an explicit time integration method. Therefore, the principal of virtual work

49


equation (using tensor notation) is [29]:

∫tA

tτij δteij dtA = t< (4.1)

where tτij is the cartesian components of the Cauchy stress tensor, δteij is the strain

tensor corresponding to virtual displacements, tA is the area at time t and t< is:

t< =

∫tA

tf bi δui d tA +

∫tBf

tfBi δuB

i d tl (4.2)

where tf bi are components of external applied forces per unit area at time t, tfB

i are

components of externally applied boundary tractions per unit length at time t, tBf

is the boundary on which external tractions are applied, δui are components of vir-

tual displacement vector and δuBi are the components of the virtual displacement

vector evaluated on the boundary tBf . The components of δui corresponding to

the prescribed displacement on the boundary tBu are zero.

As in Chapter 3, the left hand side of the equation is the internal virtual work

and the right hand side is the external virtual work ,but the configuration at time t

is used. The strain tensor is given by:

δteij =1

2

(∂δui

∂ txj

+∂δuj

∂ txi

)(4.3)

where txj are the cartesian coordinates of material points at time t and j = 1, 2. It

can be seen that the components of the virtual displacement vector δui are func-

tions of txj and the strain tensor is similar to the infinitesimal strain tensor in the

linear analysis given in Eq. 3.2 except that the derivatives of the displacements are

50


with respect to the current coordinates. In the nonlinear analysis, the configuration

is continuously changing which requires an incremental analysis and using appro-

priate strain and stress measures and constitutive relations that will be discussed

in Sec. 4.2 and 4.3.

The large deformation stress and strain measures used in this thesis are defined

in terms of the infinitesimal strain and the Cauchy stress measures later in Eqs. 4.21

and 4.19. Using these definitions, the following equality for the left hand side of

Eq. 4.1 can be obtained [29]:

∫tA

tτij δteij dtA =

∫rA

trSij δ t

rεij drA (4.4)

where trSij is the second Piola-Kirchhoff stress tensor and t

rεij is the Green-Lagrange

strain tensor. The left superscript t indicates at which configuration the quan-

tity occurs whereas the left subscript r indicates the reference configuration that

the quantity is measured with respect to. The reference configuration can be any

known configuration that was previously calculated from time 0 to time t. For the

total Lagrangian (TL) formulation, the original configuration at time 0 is used as a

reference. Therefore, for the TL formulation, Eq. 4.1 can be rewritten using Eq. 4.4

as follows: ∫0A

t0Sij δ t

0εij drA = t< (4.5)

Using Eq. 4.19, the right hand side of Eq. 4.5 can be written as follows [29]:

∫0A

t0Sij δ0eij d0A = t< (4.6)

51


The displacements of the material particles are the only variables in the equa-

tion of motion. In order to discretize the equation of motion given in Eq. 4.6 and

obtain the governing FE equations, an analogy that is similar to the linear analysis

given in Chapter 3 is employed. The body is divided into finite elements and the

nodal displacements are solved for. Interpolation functions are used to interpolate

the element displacements and coordinates.

Therefore, by using the element coordinate and displacement interpolations,

Eq. 4.5 for a dynamic system can be written in a matrix form as follows:

M tU + D tU + t0F = tR (4.7)

where M and D are the mass and damping matrices and are calculated based on

the original configuration using Eqs. 3.18 and 3.19. tU, tU, and tU are the vectors

of nodal displacements, velocities and accelerations, respectively. t0F is the nodal

reaction forces vector and is given by:

t0F =

∫0A

t0BT

Ft0S d0A (4.8)

where t0BF is the full strain-displacement matrix , t

0S is the second Piola-Kirchhoff

stress vector, and tR is the external force vector. Based on the definition of external

work given in Eq. 4.2, tR can be written as:

tR =

∫0A

HT t0fb d 0A +

∫0Bf

HBT t0fB d tl (4.9)

where H is the interpolation function matrix, t0fb is the body forces vector, and t

0fB

52


is the boundary forces vector.

It should be noted that damping effects can be modeled in the constitutive

equation when a strain-rate dependent material law is employed. However, since

a hyperelastic material law that is not strain-rate dependant is considered in this

thesis, damping effects are incorporated directly in the dynamic equation of mo-

tion in 4.7.

4.2 The Deformation Gradient and Large Deformation

Stress and Strain Measures

In this section definitions of the strain and stress measures employed in the non-

linear analysis are given. These definitions start with the deformation gradient

followed by the Green-Lagrange strain tensor as a long deformation strain mea-

sure. Finally the Second Piola-Kirchhoff stress tensor is introduced.

4.2.1 Deformation Gradient

To deal with a continuously changing configuration in a large deformation analy-

sis, appropriate stress and strain measures must be developed . To this end, first

the deformation gradient which relates the initial configuration at time 0 before the

application of external loads to the deformed configuration at time t is defined as:

t0X =

∂tx∂0x

∂tx∂0y

∂ty∂0x

∂ty∂0y

(4.10)

53


where x and y are the components of the two-dimensional position vector p=[x y]T

of each particle in the body. Therefore if d0p describes an infinitesimal fiber of

material in the original configuration, the same fiber at time t dtp is given by:

dtp = t0X d0p (4.11)

The deformation Gradient describes the stretches and rotations that the mate-

rial undergo from time 0 to t and can be decomposed into two matrices as follows:

t0X = t

0R t0U (4.12)

where t0R is an orthogonal rotation matrix that represents rigid body rotations

and t0U is a symmetric right stretch matrix that represents the stretches that the

body has undergone. The eigenvalues of the right stretch matrix are the principal

stretches and its eigenvectors are the principal stretch directions. These quanti-

ties will be used in the definition of the constitutive model and calculations of the

stress vector that are explained in the next section.

4.2.2 Green-Lagrange Strain Tensor

The strain tensor is a measure of body deformation that is independent of rigid

body motions. It indicates how much a length of a material has changed when go-

ing from the original configuration to the deformed configuration. The large defor-

mation strain measure employed with the TL formulation is the Green-Lagrange

strain tensor that is given by:

t0ε =

1

2(t0C− I) (4.13)

54


where t0C is the right Cauchy-Green deformation tensor which is defined in terms

of the deformation gradient as follows:

t0C = t

0XT t0X (4.14)

Using the polar decomposition of the deformation gradient given in Eq. 4.12,

the right Cauchy-Green deformation tensor can be writhen in terms of the stretch

tensor, i.e.

t0C = t

0U t0RT t

0R t0U =t

0 U2 (4.15)

which indicates that the eigenvalues of t0C are the square of the principal stretches

and its eigenvectors are the principal directions.

The Green-Lagrange strain tensor can be compared with the small deformation

tensor t0e by writing it in terms of displacements using the definition of deforma-

tion gradient in Eq. 4.10 and the fact that:

tpi = 0pi + tui (4.16)

where tpi and 0pi are the current and original position vectors respectively, and

tui is the corresponding deformation vector. The elements of the Green-Lagrange

strain tensor are given by:

t0εij =

1

2

(t0Xki

t0Xkj − δij

)

=1

2

((δki +

∂tuk

∂0pi

)(δkj +

∂tuk

∂0pj

)− δij

)

=1

2

(∂tui

∂0pj

+∂tuj

∂0pi

+∂tuk

∂0pi

∂tuk

∂0pj

)(4.17)

55


where δij is the kroneckor delta. It can be seen that all the derivatives are per-

formed with respect to the original configuration and that Eq. 4.17 contains a

quadratic term which makes the analysis nonlinear compared to the small defor-

mation strain tensor which is a linear function of the deformation field,

t0eij =

1

2

(∂tui

∂ 0pj

+∂tuj

∂ 0pi

)(4.18)

Based on the definitions of the Green-Lagrange strain tensor and small strain

tensor in Eqs. 4.17 and 4.18, a variation in the current Green-Lagrange strain δt0ε is

related to the variation in the small strain tensor δte by:

δt0εij =

∂tpm

∂0pi

∂tpn

∂0pj

δtemn (4.19)

This relation was used in Sec. 4.1 in order to derive the equality given in 4.4.

4.2.3 Second Piola-Kirchhoff Stress Tensor

The second Piola-Kirchhoff stress tensor t0S is the stress measure that is used in

conjunction with the Green-Lagrange strain tensor. The relation between this long

deformation stress tensor and the Cauchy stress tensor tτ that is defined as the

force per unit deformed area is given by the following equation [29]:

tτ =tρ0ρ

t0X t

0S t0XT (4.20)

or in component form:

tτmn =tρ0ρ

∂tpm

∂0pi

∂tpn

∂0pj

t0Sij (4.21)

56


where tρ is the mass density at time t andtρ0ρ

= det t0X = tJ which is the third

invariant of the deformation gradient. Eq. 4.21 was used in Sec. 4.1 in order to

derive the equality given in Eq. 4.4.

The second Piola-Kirchhoff tensor is a symmetric tensor that maps the force to

the initial configuration of undeformed area. In case of hyperelastic models, which

are employed in this thesis, the second PiolaKirchhoff stress can be evaluated as

a derivative of an energy function with respect to the Green-Lagrange strain ten-

sor [29] as will be shown in the next section.

4.3 Ogden-based Hyperelastic Constitutive Model

The constitutive equation enters the virtual work equation in the calculation of

stresses to relate material stress to its strain. The TL formulation and the stress and

strain measures covered in the last two sections include large deformation related

nonlinear effects. However, in order to obtain a stress-strain relationship that can

properly model the material, a specific constitutive relation should be employed.

Appropriate constitutive models that account for the nonlinear properties of the

soft-tissue need to be employed in order to have an accurate evaluation of the

stresses based on strains in the body. This is essential for precise prediction of

forces and deformation within the organ. The material in this section is based

on [29, 39, 41, 42].

The hyperelastic properties that are found in soft-tissues are usually modeled

using the Ogden energy function [35, 39]. This energy function is a generalization

of the neo-Hookean and Mooney-Rivlin energy functions which are commonly

57


used in the literature. The Ogden material description is defined in terms of mate-

rial constants and the principal stretches (a deformation measure) as follows:

t0W =

∑n

µn

αn

(λαn1 + λαn

2 − 2) ≥ 0 where λ1λ2 = 1 (4.22)

where λi are the principal stretches which are the eigenvalues of the right stretch

tensor as explained in Sec. 4.2 and will be illustrated in Eq. 6.24; µn and αn are

material constants.

The Ogden energy function reduces to the neo-Hookean (for n=1 and α1=2)and

Mooney-Rivlin (for n=2, α1=2 and α1=-2) energy functions. The extra flexibility

in choosing α and n in the Ogden model can potentially provide more accurate

results. However, choosing n to be higher than 1 can cause complexity in the cal-

culations and difficulty in fitting the material constants from experiments. It can be

seen from Eq. 4.22 that the energy in the body is equal to zero when there is no de-

formation, i.e. when the principal stretches are equal to 1. The principal stretches

are higher than 1 in the case of tension and below 1 in the case of compression.

Also, the function is always positive which is a requirement for an energy func-

tion.

The Ogden function given in Eq. 4.22 is for two-dimensional planar analysis.

The constrain on the principal stretches is to enforce a totally incompressible prop-

erty since the term λ1λ2 is the determinant of the deformation gradient that de-

fines the ratio of the deformed area over the initial area. Therefore, a ratio equal

to 1 indicates that no change in area has occurred while deforming. Soft-tissues

58


are considered a type of a rubber-like material that behave as almost incompress-

ible [29, 39]. Therefore, a better assumption is only an almost incompressible ma-

terial that is modeled by modifying the stain energy function given in Eq. 4.22 to

include the constrain as a penalty term with the bulk modulus κ as a Lagrange

multiplier (hydrostatic work term) as follows:

t0W =

∑n

µn

αn

(λαn

1 + λαn2

(λ1λ2)αn2

− 2

)+

1

2κ(t

0J3 − 1)2 (4.23)

where t0J3 is the third reduced invariant of the right Cauchy-Green deformation

tensor t0C and is defined as:

t0J3 = (t

0I3)12 = (dett0C)

12 = (λ2

1λ22)

12 = λ1λ2 (4.24)

It should also be pointed out that the principal stretches λi are replaced with

λi(λ1λ2)−1/2 in order to make the terms under the summation unaffected by volu-

metric deformations. The bulk modulus κ should be several thousands times the

shear modulus in order to model an almost incompressible behavior [29].

In this thesis, the Ogden strain energy function with n=1 is employed as it

was done in [13]. However, the almost incompressibility property is enforced in

this thesis (as opposed to [13]) by modifying the Qgden energy function through

adding a hydrostatic work term that penalizes the change in area. Therefore the

employed Ogden-based energy function is:

t0W =

µ1

α1

(λα1

1 + λα12

(λ1λ2)α12

− 2

)+

1

2κ(t

0J3 − 1)2 (4.25)

59


Using the definition of the initial shear modulus in terms of these material con-

stants,

µ =1

2

n∑i=1

αiµi (4.26)

µ1 can be expressed in terms of the initial shear modulus as follows:

µ1 =2µ

α1

(4.27)

By substituting Eq. 4.27 into Eq. 4.25, the following Ogden energy function is

obtained:

t0W =

2µ

α21

(λα1

1 + λα12

(λ1λ2)α12

− 2

)+

1

2κ(t

0J3 − 1)2 (4.28)

This energy function is used to calculate the second Piola-Kirchhoff stresses by

evaluating its derivative with respect to the Green-Lagrange strain tensor:

t0Sij =

∂W

∂t0εij

(4.29)

It is also possible to find the derivatives of the energy function with respect to the

principal stretches λi using the chain rule which yields the principal second Piola-

Kirchhoff stresses, i.e.:

t0Si =

∂W

∂t0εi

=∂W

∂t0λi

∂t0λi

∂t0εi

=1

t0λi

∂W

∂t0λi

(4.30)

since t0εi = 1

2(t0λ

2i − 1). It should be noted that the right subscript i in Eq. 4.30

denotes the ith principal stress value.

By using the Ogden-based energy function given in Eq. 4.28 in Eq. 4.30, the

60


principal second Piola-Kirchhoff stresses are:

t0S1 =

1t0λ1

∂W

∂t0λ1

=2µ

λ1α2

(α

2λ−α2

1 λ−α2

2 − α

2λ−3α

21 λ

α22

)+ κ(λ1λ2 − 1)

λ2

λ1

(4.31)

t0S2 =

1t0λ2

∂W

∂t0λ2

=2µ

λ2α2

(−α

2λ

α21 λ

−3α2

2 +α

2λ−α2

1 λ−α2

2

)+ κ(λ1λ2 − 1)

λ1

λ2

(4.32)

In summary, the known values of the principal stretches for the configuration at

time t are used to calculate the principal second Piola-Kirchhoff stress values which

in turn are employed to compute the nodal reaction force vector by Eq. 4.8. Also, as

it was mentioned in the beginning of the chapter, the central difference method is

utilized to solve the governing equations given in Eq. 4.7 and find the displacement

field at time t +4t using Eq. 3.27 as explained in Sec 3.2.

61

Chapter 5

Modeling of Soft-tissue Cutting

Soft-tissue cutting is one of the most complicated surgical tasks to model. It changes

the way the elements are joined in the mesh. This is known as a topological mod-

ification. Many techniques have been used to model topological modifications as

discussed in Chapter 2. An element separation technique is developed in this the-

sis that does not increase the number of elements in the model, preserves the mass

and area of the model, and creates a cut that is closest to the path traversed by

the tool using node snapping. The cutting technique that is presented here uses

the Ogden-based hyperelastic nonlinear mathematical model with the almost in-

compressible condition that was presented in the last chapter in order to take into

account material and geometrical nonlinearities of soft-tissues.

A progressive cutting algorithm is developed by allowing the mesh to deform

until a cutting criterion, that will be introduced later in this chapter, is met. There-

fore, at each time step, the cutting criterion is checked; when the condition is met,

a cut is introduced to the FE mesh through the following steps that are also shown

in Fig. 5.1:

62


Separated Elementsand Snapped

Node Determination

Local Remeshing

Update original matrices andarea of local elements

Generate new node andupdate vertices matrix

Mesh relaxationsteps

Apply load oncut node as a

new loading node

Ele

ments

Separa

tion

and N

ode S

nappin

g

Update cut node position andlocal elements vertices

Mesh U

pdate

sF

orc

e F

eedback

Calc

ula

tions

Pe

rfo

rm c

utt

ing

Iscutting criterion

met?

Yes

NoDetermine nodaldeformations to

evaluate current position

Increment tool tipposition for thenext time step

Figure 5.1: Flow chart of the algorithm employed to perform cutting

63


• Elements separation and node snapping: Upon meeting the cutting crite-

rion, the next node that is closest to the cut path is determined and snapped

to the path. This is achieved by projecting the node into the cut path in order

to have a cut that best fits the tool direction. In some cases, node snapping

could generate badly shaped elements or elements with very small area that

could cause numerical instability in the simulation. To avoid that, the lo-

cal cutting area is remeshed by retriangulating the elements and allowing

the snapped node to move if necessary in order to keep the element’s edge

lengths bigger than a threshold value that is specified by the time step. This

will eliminate the necessity of decreasing the time step when the element’s

area becomes smaller than a threshold value.

• Mesh updates: After performing the local remeshing, the original matrices

and areas of the remeshed elements are updated using an inverse parametric

method that is based on shape functions in order to find the original position

of the snapped node. The cut is introduced by splitting the load node into

two nodes with each one having half the mass of the original node and the

same original and current position. Also, the nodal reaction forces between

the separated elements disappear by updating the triangle’s vertices in or-

der to make the elements on one side of the cut path have the original node

number and the elements on the other side have the new node number.

• Force feedback calculations: To generate a haptic feedback that is similar to

what is obtained in experiments shown in the literature [1,66], few time steps

are needed for mesh relaxation after node snapping. The mesh relaxation is

performed locally by letting the mesh deform without an external force on

64


the load node. This step allows the forces to decrease after a cut is performed

and then the forces start to increase again in the following time steps when

the load is applied on the snapped node. In order to avoid overlapping of

the nodes on the cut path, a constrain on these nodes’ position is added by

forcing the nodes to stay on one side of the cut path. The tool is assumed to

have a known trajectory.

The following sections include a detailed explanation of each step of the algo-

rithm briefly described above.

5.1 Cutting Criterion

To determine when a cut should be created in the soft-tissue model, the external

force exerted on the the tissue model is used as a cutting criterion. A similar ap-

proach has been employed in [66, 67]. Determining the threshold value for cutting

is straight forward with this approach since soft-tissue cutting experiments pre-

sented in the literature [1,66] measure the force on the tip of the tool. If the external

force exceeds the maximum load that the soft-tissue can resist then a cut should be

executed. This maximum external force is dependent on the material property that

would indicate its toughness. Also, it is possible to make the critical external force

value dependent on the cutting tool sharpness, tool velocity and previous damage

in the cutting area as it was explained in Sec. 2.3.

After calculating the external force on the contact node between the tool tip

and the FE mesh (load node) in each time step, this force is decomposed into two

components relative to the tool which is modeled as a line since a two-dimensional

65


analysis is performed. One component is along the tool direction, whereas the sec-

ond component is perpendicular to this direction. The first is used as a physical cri-

terion for cutting. Therefore, when the external force in the tool direction reaches a

threshold value, the modeled object should be broken apart and the cutting proce-

dure is performed by carrying out the following steps in order to separate elements

and change the topology of the FE mesh.

5.2 Elements Separation and Node Snapping

5.2.1 Determination of Separated Elements and Snapped Node

Since element separation technique is used to create the cut in the FE mesh, the two

triangular elements (cut elements) that must be separated should be determined

first. These cut elements are the two elements that share the cut edge between

the current contact node (load node) where the load is being applied and the next

contact point at which the load will be applied after the cut is performed. The

second node of the cut edge (cut node) is chosen based on the angle that the cut

edge makes with the cutting tool trajectory. Therefore, the edge that makes the

smallest angle with the cutting tool trajectory is chosen as the cut edge and the

two elements that share this edge are the cut elements that will be separated. This

process is illustrated in Fig. 5.2 where θ1 < θ2. Boundary edges are excluded from

the possibility of being a cut edge in order to avoid changing the overall area of

the object when node snapping is performed.

In order to have a smooth cut is as close as possible to the cutting trajectory, the

66


θ1 θ2

Cutting Trajectory

Cut element 1

Cut element 2Cut edge

Cut node

Load nodeSnapped node

A B

Cutting tool

Figure 5.2: Illustration of determination of separated elements and node snapping:(A) Before node snapping. (B) After node snapping

second node on the cut edge (cut node) should be snapped to the line that repre-

sents the tool trajectory. This is achieved by determining the orthogonal projection

of the cut node on the cutting trajectory and moving the cut node to that point (see

Fig. 5.2). It should be noted that the cut node is the node where the load will be

applied after the cut. Therefore the tool tip position should be offset to the orthog-

onal projection point where the cut node is placed in order to apply the load on

this node in the following time steps.

5.2.2 Local Remeshing

Node snapping that is performed in the previous step could generate badly shaped

elements or elements with very small area which are called degenerated trian-

gles as shown later in Figs. 5.5 , 5.6, and 5.7. These elements can cause numeri-

cal instability in the simulation because of their small characteristic length which

would make the critical time step for guaranteed lower than the employed time

67


step [14, 29]. Therefore, to avoid the necessity of decreasing the employed time

step due to degenerated elements, the local cutting area which includes the ele-

ments that surround the snapped node is remeshed. The remeshing algorithm re-

triangulates the elements and allows the snapped node to move if needed without

creating any new elements in the mesh. The local remeshing algorithm employed

in this thesis is a modification of the meshing algorithm given in [71] and is shown

in Fig. 5.3. The area that is remeshed is a polygon specified by the nodes of the

elements in contact with the snapped cut node. The boundary nodes of these ele-

ments are fixed. The steps of the algorithm includes the following:

1. The area specified by the nodes is triangulated using the Delaunay function

which returns a set of triangles such that no data points are contained in any

triangle’s circumscribed circle [71]. The Delaunay triangulation maximizes

the minimum angle of all the angles of the triangles in the triangulation.

2. The quality of the triangular elements created are calculated as a measure of

the size and shape of the elements. This is employed to detect the degener-

ated elements (e.g. see Fig 5.4) that can cause numerical instability because

of small edge length or small area. As it was mentioned before, the character-

istic length of the element determines the critical time step. Also, as the area

of the element tends to zero, force computations give infinite deformations

that indicate numerical instability [49]. Many quality measures are available

in the literature [72]. The quality measure employed in this thesis is:

quality(element) = 4√

34

(L21 + L2

2 + L23)

(5.1)

68


Exit

Triangulate the nodes usingDelaunay function

Calculate the quality ofgenerated elements

Use positioncalculated by forces

Retriangulate elementsand calculate quality

Find projection of thispoint on the cut line

Use projection point ascut node position

Obtain nodes positionto specify the

remeshing area

Ismin. Quality

improved?

Ismin. Quality

> Thresh.?

Find cut node positionby force-displacement

function

Yes

Yes

No

No

Local remeshing

Separated Elementsand Snapped

Node Determination

Update cut nodeposition and localelements vertices

Figure 5.3: Flow chart of the local remeshing algorithm

69


A B

Figure 5.4: Examples of degenerated triangular elements: (A) Small edge length.(B) Small area.

where 4 is the area of the triangular element and Li is the length of each

edge in the triangle. The constant is employed to normalize the measure

for equilateral triangles and makes the measure take on values between zero

for complete degenerated elements and one equilateral elements. Therefore,

degenerated elements that can cause numerical instability have low quality

measure.

3. The calculated quality of each element is compared with a threshold value

that is known to satisfy the stability condition of the explicit integration method.

If the quality of all elements are above the threshold value then the algorithm

exits with the triangles obtained from the Delaunay function which can be

a retriangulation of the elements in the original mesh as shown in Fig. 5.5.

Otherwise, the internal node, which is the snapped cut node, is repositioned

using a linear force-displacement relationship to solve for an equilibrium in

a truss structure [71] as follows:

pn+1 = pn +4tE F(pn) (5.2)

where pn+1 and pn is the current and previous position of the snapped node

and 4tE is the time step in Eulers method. F(pn) is the force excreted on

the snapped node based on the previous position and it is the sum of the

70


Cut node

Cutting tool

Snapped cut node

Degenerated element

Load node

Re-triangulatedelements

A B C

Figure 5.5: Example to illustrate local remeshing with retriangulation: (A) Beforenode snapping. (B) Degenerated elements creation after node snapping. (C) Retri-angulation of local elements.

repulsive force vectors from all edges connected to the node:

F(pn) =∑

edge

fedge (5.3)

The repulsive force of each edge fedge depends on the edge current length `

and the unextended length `0 as follows:

fedge(`, `0) =

k(`0 − `) if ` < `0

0 if ` ≥ `0

(5.4)

Therefore, the resultant force F(pn) moves the snapped node in order to

equalize the length of the edges connected to the snapped node after few

iterations. This process will elongate the short edges which improves the

quality of the degenerated elements as shown in Fig. 5.6.

4. In some cases, the node repositioning using the force generated based on the

length of the edges is away from the cutting tool trajectory and sometimes

71


Remeshed localelements

A B C

D E F

Cut node

Cut nodeCutting tool

Load node

Snapped cut node

Degeneratedelement

Snapped noderepositioning

Re-triangulatedelements

Figure 5.6: Example to illustrate local remeshing with snapped node repositioning:(A) Before node snapping. (B) Degenerated elements creation after node snapping.(C,D,E,F) Local elements remeshing by retriangulation and snapped node reposi-tioning.

72


moving the node away from the cut line is not necessary to improve the qual-

ity of the element as it is shown in Fig. 5.7C. Therefore, in order to have a cut

that is as close as possible to the cutting profile whenever it is possible, the or-

thogonal projection of the new position on the cutting tool trajectory is found

and it is assumed to be the new position of the cut node as shown in Fig. 5.7D.

If the projection point improves the quality of the elements, it is taken as the

new position of the snapped node; whereas, if it worsens the quality of the

elements, the position calculated using the repulsive forces as explained in

the previous step is taken as the new position of the snapped node. With this

approach, the snapped node will be placed on the cut trajectory whenever it

is possible in order to create a cut that is as close as possible to the intended

cutting path.

5.3 Mesh Updates

After remeshing the local cutting area, the following updates should be performed:

1. The position of the snapped cut node is updated with the value obtained

from the local remeshing algorithm, which will guarantee the stability of the

simulation in the subsequent time steps. The vertices of the local remeshed

elements should be updated because the Delaunay function can change the

topology of these elements by retriangulation as it was shown in Figs. 5.5

and 5.6.

2. In order to update the original matrices and areas of the remeshed elements,

the new original position of the snapped node 0pnew should be found since

73


A B

C D

Cut nodeCutting tool

Load node

Snapped cut nodeDegenerated

element

Snapped noderepositioning

Snapped noderepositioning by

projection oncutting trajectory

Figure 5.7: Illustration of local remeshing by snapped node repositioning with or-thogonal projection: (A) Before node snapping. (B) Degenerated elements creationafter node snapping. (C) Local elements remeshing by snapped node repositioning(D) Local elements remeshing by projecting the new snapped node on the cuttingtrajectory.

its new position tpnew is calculated based on the current deformed configura-

tion as it was explained above in Sec. 5.2. The inverse isoparametric mapping

method mentioned in [63, 73, 74] is employed to find the original position of

the snapped node. This method involves first finding the element that con-

tains the new point by utilizing the polygon bounding box method and then

obtaining the isoparametric coordinates (h1, h2, h3) of the new point with re-

spect to the three nodes of this element. As it was explained in 3.1.2, these

coordinates define the interpolation function of the new point and they are

74


found using the current position of the three nodes of the element that con-

tains the new point using the following three equations:

txnew = h1tx1 + h2

tx2 + h3tx3 (5.5)

tynew = h1ty1 + h2

ty2 + h3ty3 (5.6)

h1 + h2 + h3 = 1 (5.7)

where Eqs. 5.8 and 5.6 are based on Eq. 3.9 and show how the x and y co-

ordinates of a point in an element is defined based on the xi and yi coordi-

nates of the three nodes in the element using the isoparametric coordinates.

Eq. 5.7 is a property of the isoparametric coordinates. All nodes coordinates

(txi and tyi) and the new point coordinates (txnew and tynew) are known in

the deformed configuration; therefore, the isoparametric coordinates can be

found by solving Eqs.( 5.8)-( 5.7). Therefore, the new original position of the

snapped node 0pnew can be computed using the obtained isoparametric coor-

dinates and the original positions of the element’s three nodes 0pi as follows:

0pnew =3∑

i=1

hi0pi (5.8)

where p = [x y]T . The displacement field ,tU and t−4tU of the snapped node

should also be mapped from the old to the new position by employing the

inverse isoparametric mapping method as well based on Eq. 3.8.

The area4 and elemental strain-displacement matrix Bm mentioned in Eq. 6.5

75


should be recalculated for the remeshed local elements using the new orig-

inal position of the snapped cut node. The details of how these calcula-

tions are performed will be explained in 6.1.1. It should be noted that the

mass of the nodes in these elements are not updated according to the area

of the elements. This is to avoid having small masses at nodes connected

to small elements that could potential cause numerical instability. instead,

the same mass matrix that is generated based on the original configuration is

employed.

3. To separate the two elements and create a cut in FE mesh, a new node is

established by splitting the load node into two nodes with each one having

half the mass of the original node and the same original and current position.

Also, in order to eliminate the nodal reaction forces between the separated

elements, triangles’ vertices matrix is updated to make the elements on one

side of the cut trajectory have the original node number and the elements on

the other side have the new node number.

5.4 Force Feedback Calculations

Typical soft-tissue cutting experiments found in the literature [1, 66] show that the

force at tip of the surgical tool drops down when a cut is created in the tissue and

when a further loading is applied, the force starts increasing again as shown in

Fig. 5.8. In order to have a similar behavior in the modeling process and gener-

ate a haptic feedback that is comparable with experimental results reported in the

literature, the following steps are performed:

76


1. A number of time steps are needed for mesh relaxation after node snapping

and local remeshing are performed. The mesh relaxation is done locally on

the remeshed elements only. The nodes of these elements are set to deform

freely to the relaxed position without any external force applied on the load

node. Therefore the position of these nodes are updated after each time step

and a higher damping is used during these steps in order to reach the re-

laxed position without many oscillations. It should be noted that these time

steps are internal and are not used for the actual deformation and force cal-

culations. They only allow for decreasing the high forces generated after the

remeshing when a cut is performed because of the sudden big change in the

snapped node position in one time step.

2. Preparing for the next time step, a load is applied on the cut node as a new

load node. The response of the whole object is obtained by calculating the

deformation field given the applied external force.

After calculating the new position of nodes in each time step,the position of the

nodes on the cut line is examined. To avoid the overlapping of these node after a

cut, a constrain on the position of these nodes is added to force them to stay on the

same side of the cut path all the time. Thus, if it is found that a node has crossed

the cut path, then its position is changed to the orthogonal projection of the node

position on the cut path.

77


Figure 5.8: Experimental data from liver cutting. Courtesy of [1]

78

Chapter 6

Deformation and Cutting Simulation:

Algorithms and Results

This chapter discusses the implementation of the deformation/cutting modeling

algorithms and presents the results of numerical simulations to evaluate the per-

formance of the proposed modeling techniques. Based on the material of Chapter

3, the first section shows the detailed computations that need to be carried out

when a linear FE analysis is employed with the explicit time integration method.

An overview of the computations involved in the nonlinear FEM algorithm is

given in the second section, which is essentially an extension of Chapter 4. The

third section gives a summary of the overall algorithm employed to simulate suc-

cessive soft-tissue cutting based on the steps given in Chapter 5. Each of the sec-

tions also contains the results of numerical simulations conducted with the models

employed.

79


6.1 Steps and Results of 2D Explicit Dynamic Linear

Finite Element Algorithm

The linear analysis is included in this thesis for comparison the nonlinear FE model.

The steps of the algorithm for linear FEM analysis can be divided into two main

groups. The first group includes calculations that can be performed off-line prior

to the start of the simulation whereas the second group includes on-line computa-

tions that must be conducted at each time step. Fig. 6.1 shows the main blocks of

the algorithm and below are the details of what is performed at each step.

Generate meshand load

boundary conditions

Generate elementalstiffness matrices

Compute and diagonalizemass and damping

matrices

Initialize variablesand apply load

Compute elementalnodal elastic forces

Obtain net nodalelastic and

external forces

Compute nodaldisplacements

Update variablesand apply load

for next time step

Off-lin

e c

om

puta

tions

Onlin

eco

mpu

tatio

ns

Figure 6.1: Steps of 2D linear dynamic finite element algorithm

80


6.1.1 Off-line Precomputations and Initializations

The following calculations can be performed prior to the start of the simulation:

Generate the Mesh and Specify Load and Boundary Condition Nodes

The available MATLAB source code of the mesh generator presented in [71] is used

to generate the mesh of a 2D object. To this end, the geometry of the object should

be defined as a distance function that returns the signed distance from each node

position to the closest boundary. The distribution of elements in the mesh whether

uniform or non-uniform can be specified by an edge length scaling function. The

required resolution of the mesh (fine or coarse) is determined by a number that rep-

resents the initial edge length in the mesh. In this thesis, without loss of generality,

the 2D object was defined as a circle with a radius of 0.1m which is representing

a slice of a 3D deformable object. The object is uniformly meshed with triangle

elements that have an approximate edge length of 0.02m as it is shown in Fig. 6.2.

All interaction forces and deformations occur in the 2D plane.

The output of the meshing function is two matrices. The first matrix (P) speci-

fies the nodes positions. The size of this matrix is (Nx2) where N is the number of

nodes. In each row of the matrix, the first element is the x coordinate of the corre-

sponding node position (row number) and the second element is the y coordinate

of that node position. The second matrix (T) consists of the indices (nodes) of

each element. Since the triangular elements are utilized, each element has 3 nodes.

Therefore, the size of this matrix is (NEx3) where NE is the number of triangle

elements.

To prevent the object from moving freely, a sufficient number of nodes are

81


-0.1 -0.05 0 0.05 0.1 0.15

-0.1

-0.05

0

0.05

0.1

0.15

1

2

3

4

5

6

7

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Figure 6.2: Meshed circle with the designated node numbers

fixed. These nodes are called boundary condition nodes and act as a support to

the meshed object. The number of the nodes that are needed to be fixed should be

specified by the user. At the same time, the node number where the load is applied

should also be provided.

Generate the Elemental Stiffness Matrices

Since a linear elastic system is being solved, the stiffness matrix is constant through

out the simulation. Therefore, the elemental stiffness matrices can be calculated in

the precomputation stage.

As it was shown before in Eq. 3.14, the stiffness matrix for each element can be

calculated using the following integral:

Km =

∫

Am

BTmCmBmdAm (6.1)

82


where Bm is the element strain-displacement matrix, Cm is the element matrix of

material constants. Since these matrices are constant for a specific element, the in-

tegral is performed on the area of each element and the elemental stiffness matrices

are computed according to the following equation:

Km = BTmCmBm4 (6.2)

where 4 is the area of each triangular element. Therefore, in order to find the

stiffness of each element, Bm, Cm and 4 have to be calculated for each element as

follows:

• Cm calculation: Since the object is considered to have one type of material,

Cm is the same for all elements in the mesh and needs to be calculated only

once. The material matrix for an isotropic material when plane strain analysis

for a 2D slice of a unit thickness is being performed is given by [29, 56]:

Cm =E

(1 + ν)(1− 2ν)

1− ν ν 0

ν 1− ν 0

0 0 1−2ν2

(6.3)

where ν is the Poisson’s ratio and E is the Young’s modulus which are the

parameters that specify the properties of the material. A Poisson’s ratio of

ν=0.49999 that represents an almost incompressible material and a Young’s

modulus of (E=1800 Pa) are utilized in the simulation.

83


• 4 calculation: The area of each triangle is calculated as follows:

24 = det

1 x1 y1

1 x2 y2

1 x3 y3

(6.4)

where x and y are the coordinates of nodes 1, 2, and 3 of each element. There-

fore, the P and T matrices that were generated by the meshing algorithm can

be used to calculate the area for each element as well as the B matrix of each

element as explained in the next step.

• Bm matrix calculation: This matrix needs to be calculated for each element

separately because it depends on the cartesian coordinations of the nodes of

each element as it will be shown below. Based on the strain definition in

Eqs. 3.2 and 3.10, and Eq. 3.6 that shows the calculation of the displacement

field using the element nodal displacements, the Bm matrix can be computed

as follows:

Bm = LHm =

∂∂x

0

0 ∂∂y

∂∂y

∂∂x

h1 0 h2 0 h3 0

0 h1 0 h2 0 h3

(6.5)

where L is a linear operator matrix, Hm is the element shape function ma-

trix and hk is the shape function of each node of the element. For the linear

triangle element, this function is given by:

hk =ak + bkx + cky

24 (6.6)

84


where k=1, 2, and 3 correspond to the local node number for each element.

Therefore, by substituting Eq. 6.6 in Eq. 6.5, the B matrix for each element is

given by:

Bm =1

24

b1 0 b2 0 b3 0

0 c1 0 c2 0 c3

c1 b1 c2 b2 c3 b3

(6.7)

where

b1 = y2 − y3 c1 = x3 − x2

b2 = y3 − y1 c2 = x1 − x3

b3 = y2 − y2 c3 = x2 − x1

Based on Eq. 6.2, the size of the stiffness matrix generated for each element is a

6×6 given that the elements have 3 nodes and each node has 2 coordinates.

Compute, Diagonalize and Assemble Elemental Mass and Damping Matrices

The mass matrix of each element is calculated using the following integral which

was derived in 3.18:

Mm =

∫

Am

ρmHTmHmdAm (6.8)

Using the linear shape functions of the triangle given in Eq. 6.6, the left hand

side of Eq. 6.9 is obtained. In order to have a diagonal mass matrix which is needed

to simplify the computations when explicit time integration method is employed,

the row sum method is used [56]. In this method, the elements of each row are

added and the sum is placed in the diagonal entry of that row. The lumped mass

85


matrix is shown in the right hand side of the following equation:

Mm =M

12

2 0 1 0 1 0

0 2 0 1 0 1

1 0 2 0 1 0

0 1 0 2 0 1

1 0 1 0 2 0

0 1 0 1 0 2

Diagonalize−−−−−−−−−−→Row sum method

Mm =M

3

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

(6.9)

where M=total mass of the element = ρm4, ρm= mass density of the element and

all elements are assumed to have the same mass density, and 4=area of each tri-

angular element.

To generate the system mass matrix, the elemental mass matrices can be assem-

bled by adding the contribution of each element to generate the mass at each node

as follows:

M =∑m

Mm (6.10)

The damping matrix D is generally not assembled from the element damping

matrices as is given in 3.19 because of the difficulty of determining the element

damping parameter. Instead, it is usually calculated as a function of the global

stiffness and mass matrices. In this thesis, the damping matrix is calculated as a

linear function of the global mass matrix as follows:

D =2α

4tM (6.11)

86


where α is a scaling factor. This ensures that the global damping matrix is diag-

onal which is needed for elemental level computations to avoid complex matrix

inversion in explicit integration of the system dynamics.

Initialize Displacement Field and Apply load

Since a dynamic analysis is performed, initial conditions need to be taken into ac-

count. Also, as it can be seen in Eq. 3.27, the calculation of the displacement vector

at the first time step 4tU requires the displacement vectors from the previous two

steps 0U, −4tU. The displacement variables are initialized assuming that the at

time 0 the object is at rest which means that the nodal displacements vectors are

zero at and prior to time step zero, i.e.

−4tU = 0, 0U = 0 (6.12)

The load is specified as a displacement constraint on the contact node where

the load node displacement for the first time step is given by:

4tUload = [utool, vtool] (6.13)

where utool is the contact node displacement in the x direction, and vtool is the con-

tact node displacement in the y direction.

6.1.2 Time Stepping

Given all the matrices that have been calculated in the off-line precomputation

stage above, in each time step the following calculations are performed in order to

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find the 2D object response to the applied load:

• Compute the nodal elastic forces for each element as is given in Eq. 3.26 using

elemental stiffness matrix and elemental nodal displacement vector that are

known at time t

tFm = Km.tum (6.14)

• Since nodes are usually shared by a few elements, the contribution of each

element should be summed in order to obtain the net nodal elastic forces at

time t, tF:

F =t∑m

Fm

• Obtain the net nodal external forces vector at time t, tR which is zero at all

the nodes of the mesh except the nodes at which external contact occurs.

The nodes where an external forces exist and should be calculated include

the load node and the boundary condition nodes. Equation 3.27 is used to

calculate the external force on these nodes since the displacement at time

t +4t at these nodes are known in Eq. 6.13. Therefore, for the load node the

external force is

tRload =

(Mload

4t2+

Dload

24t

)t+4tUload +

(Mload

4t2− Dload

24t

)t−4tUload

+ Fload −(

2Mload

4t2

)tUload (6.15)

• Compute displacements using Eq. 3.27 that was derived based on the central

difference method

88


• Update displacements to be used in the next time step:

t−4tU =t U , tU =t+4t U (6.16)

and apply load for the next time step by increasing the displacement field on

the load node which was specified by Eq. 6.13 above.

6.1.3 Simulation Results

To show the results of using the algorithm explained above to simulate the de-

formations in a soft object, the generated mesh shown in Fig. 6.2 is employed.

Tabel 6.1 summarizes the parameters utilized in the linear analysis which are ex-

plained throughout the section.

Parameter ValueRadius of the simulated body 0.1 mApproximate Element edge length 0.02 mPoisson’s ratio (ν) 0.499Young’s modulus (E) 1800 PaMass density (ρ) 1000 kg/m3

Time step (4t) 0.001 sDamping scaling factor α 0.05

Table 6.1: Simulation parameters employed in linear FE analysis

The time step employed guarantees a stable system given the size of elements in

the employed mesh. Based on the relationship given in Eq. 6.11 and the damping

scaling factor utilized, each node in the mesh has a damping of 100 times its mass.

In the simulations, Nodes 31, 42, and 43 are the fixed boundary condition nodes

and the load is applied at Node 41. These nodes are specified by the user through

the graphical user interface shown in Fig. 6.3.

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Figure 6.3: Graphical user interface to specify loading and boundary conditionnodes

At each time step, the tool tip position (Node 41 position) is displaced by a

loading step of 0 in the x axis direction and -0.0001 in the y axis direction. Fig. 6.4

shows a sequence of the deformed mesh at different stages. The first stage shows

the mesh before deformation when no load is applied on the mesh. The second

stage displays the deformed mesh after 200 time steps. The third figure is the de-

formed mesh after 400 time steps, and finally the last part is a comparison between

the first and the third parts. The progress in the tool tip position over the time is

shown as a think line.

To illustrate the linearity of the analysis, in Fig. 6.5, the deformations vs. ex-

ternal force profile in the y axis direction on Node 41 over 400 time steps for two

different loading steps are compared. The first is for a loading step of -0.0001 in y

axis direction and the second is for a loading step of -0.00005 in y axis direction. It

can be seen that when the loading step is doubled, the external forces are doubled

as well which is a property of a linear system.

90


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A B

C D

Figure 6.4: Simulation of soft-tissue deformation using linear analysis: (A) Originalmesh. (B) Deformed mesh after 200 time steps. (C) Deformed mesh after 400 timesteps. (D) Comparison between A and C.

91


-0.05 -0.04 -0.03 -0.02 -0.01 0-90

-80

-70

-60

-50

-40

-30

-20

-10

0

X: -0.01995Y: -40.6

Displacement

Exte

rna

l fo

rce

X: -0.0399Y: -81.2

Loading step = -0.0001


Figure 6.5: Displacement vs. external force profile at Node 41 in the Y axis directionfor linear FE analysis

92


6.2 Steps and Results of 2D Explicit Dynamic Nonlin-

ear Finite Element Algorithm

In general, the algorithm employed for the nonlinear analysis has the same steps

as the linear analysis mentioned in the previous section and shown in Fig. 6.1. One

of the main differences between the algorithms for the two types of analysis is cal-

culation of the nodal elastic forces. In the nonlinear analysis, there is no need to

calculate the elemental stiffness matrices since the elemental elastic force vectors

are not calculated using the elemental stiffness matrices and elemental nodal dis-

placement vectors as it given in Eq. 6.14. Instead the elastic forces are calculated

using the following integral as it was given in Chapter 4:

∑e

t0Fm =

∑e

∫0Am

t0BT

Fm

t0Smd0Am (6.17)

where t0BT

Fmis the elemental full strain-displacement matrix at at time t. Therefore,

the spatial derivatives of the interpolation functions which are computed with re-

spect to the initial configuration can be calculated off-line using Eq. 6.7 as it was

explained in 6.1.1. The steps of computing the elemental nodal elastic forces for

the nonlinear analysis are detailed below.

6.2.1 Elemental Nodal Elastic Force Vectors Calculation

To calculate the nodal elastic force vector for each element, the steps shown in

Fig. 6.6 are performed for each element in the mesh. These steps are explained

below.

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Deformation GradientCalculation

Full Strain-displacementMatrix Calculation

Stress VectorCalculation

Elastic Force VectorCalculation

Principal StressCalculation

Figure 6.6: Steps to calculate nodal elastic force for each element

Calculation of the Deformation Gradient

The deformation gradient for a three node constant strain triangular element can

be derived based on Eq. 4.10 and using the shape function given in Eq. 6.6. As it

was given in Eq. 3.9, the shape function is employed to define the position of any

point within the element in terms of the element nodes position as follows:

tp =3∑

k=1

hktpk (6.18)

where tp = [tx ty]T since a two-dimensional analysis is performed. Each compo-

nent of the deformation gradient matrix can be calculated using the derivative of

94


the shape function with respect to the initial position of the nodes, i.e.

∂tpi

∂tpj

=3∑

k=1

(∂hk

∂0pj

)tpk

i (6.19)

Using the definition of the shape functions in Eq. 6.6, their derivatives with respect

to the initial position of the nodes are

∂h1

∂0x=

b1

24 ;∂h1

∂0y=

c1

24∂h2

∂0x=

b2

24 ;∂h2

∂0y=

c2

24 (6.20)

∂h3

∂0x=

b3

24 ;∂h3

∂0y=

c3

24

where bk and ck are calculated as it was shown for Eq. 6.7 using the nodes positions

in the initial configuration at time 0, and 4 is the area of each element in the ini-

tial cofiguration. Therefore, using Eq. 6.19 and Eq. 6.20, the deformation gradient

matrix for each element can be written as:

t0X =

∂tx∂0x

∂tx∂0y

∂ty∂0x

∂ty∂0y

=

1

24

b1tx1 + b2

tx2 + b3tx3 c1

tx1 + c2tx2 + c3

tx3

b1ty1 + b2

ty2 + b3ty3 c1

ty1 + c2ty2 + c3

ty3

(6.21)

A different method for deriving the deformation gradient for a constant strain

triangular element is given in [57] which would produce the same results as those

obtained here.

95


Full Strain-displacement Matrix Calculation

The strain-displacement matrix in the nonlinear analysis is not constant as it was

the case in the linear analysis in the previous section. The full strain-displacement

matrix has to be recalculated for each element at every time step based on the

constant strain-displacement matrix that is calculated off-line and the current de-

formation gradient as follows:

t0B(k)

F =t0 B(k)

0t0XT (6.22)

where t0B(k) ∈ R3×2 are the two columns of the t

0B that corresponds to node k

and t0BF ∈ R3×6. The method for calculating t

0BF given in [29] produces the same

results.

Second Piola-Kirchhoff Stress Vector Calculation

The second Piola-Kirchhoff Stress measure can be defined as symmetric matrix or

a vector,

t0S =

t0S11

t0S12

t0S21

t0S22

; t

0S =

t0S11

t0S22

t0S12

(6.23)

The values of this measure are calculated for each element at every time step as

follows:

1. Calculate the right Cauchy-Green deformation tensor t0C using the deforma-

tion gradient given in Eq. 4.14.

96


2. Perform the eigenvalue decomposition for the right Cauchy-Green defor-

mation tensor in order to obtain the eigenvectors that represent the princi-

pal directions and the eigenvalues that represent the square of the principal

stretches as explained in Sec. 4.2. The decomposition is given by:

t0C =

[V p1 V p2

]

λ21 0

0 λ22

[V p1 V p2

]−1

(6.24)

where λ1 and λ2 are the principal stretches; V p1 and V p2 are the principal

directions.

3. Use the principal stretches to calculate the principal values of the second

Piola-Kirchhoff stress tensor according to Eq. 4.31 and Eq. 4.32. Many values

for the bulk modulus κ where tested and it was found that a value higher

than 104 results in a stable simulation while enforcing the incompressibility

constrain. The higher the value of κ is the better the constrain is imposed;

however, very large values could lead to very high forces generated through

the calculations. Therefore, a value of 2 × 105 is employed during the simu-

lations.

4. The second Piola-Kirchhoff stress tensor is calculated using its eigenvectors

and eigenvalues. The principal stresses calculated in the previous step are

the eigenvalues of the second Piola-Kirchhoff stress matrix. For isotropic hy-

perelastic materials, the principal axes for the stress tensor coincide with the

principal axes of the strain tensor [75, 76]. Therefore, the principal directions

97


of the second Piola-Kirchhoff stress tensor are the same as the principal direc-

tions of the right Cauchy-Green deformation tensor t0C that are obtained in

Step 2 above. These eigenvectors are at the same time the principal directions

of the right stretch matrix t0U as it was explained in Chapter 4. The second

Piola-Kirchhoff stress vector that is used next is obtained by rearranging the

components of the matrix and taking one of the off-diagonal components

since the matrix is symmetric.

Elastic Force Vector Calculation

The elastic force vector for each element is computed based on the integral given

in Eq. 6.17. The obtained full strain-displacement matrix in Eq. 6.22 and the second

Piola-Kirchhoff stress vector are used to calculate the elastic force vector along with

the area of each element as follows:

t0Fm =t

0 BTFm

t0Sm4 (6.25)

Finally, these elemental elastic force vectors are assembled in order to obtain the

global force vector that is used to evaluate the displacements using Eq. 3.27 as

explained in Sec. 6.1.

6.2.2 Simulation Results

Simulations were carried out using the same mesh and loading conditions as in the

case of the linear analysis in Sec. 6.1.3. Also, the parameters employed in the non-

linear analysis are the same as the ones summarized in Table 6.1 with the addition

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of few other parameters summarized in Table 6.2.

Parameter ValueShear modulus µ 600 Pa [13]Material constant α -4.7 [13]Bulk modulus κ 2× 105 Pa

Table 6.2: Simulation parameters employed in nonlinear FE analysis

Fig. 6.7 shows a sequence of the deformed mesh at different stages by employ-

ing the algorithm for the nonlinear analysis. The first stage shows the original

mesh. The second stage shows the deformed mesh after 200 time steps. The third

figure is the deformed mesh after 400 time steps, and finally the last part is a com-

parison between the first and the third stages of the simulation. On the other side,

Fig. 6.8 shows the state of the mesh when the load is released from the deformed

configuration obtained earlier. It shows that the mesh relaxes back to the origi-

nal undeformed configuration after few hundred time steps. The number of time

steps required to get to the original configuration depends on the damping scaling

factor employed.

The results obtained from the linear and nonlinear analysis are compared in

Fig. 6.9 where the mesh deformations for the time step 400 are displayed together.

It can be seen and it was also verified that the incompressibility constrain was en-

forced in the nonlinear analysis and the area of the mesh was kept almost constant

before and after the deformations. The nonlinearity of the analysis is illustrated in

Fig. 6.10 where the deformations vs. external force profile in the y axis direction on

Node 41 over 400 time steps for two different loading steps (-0.0001 and -0.00005)

are displayed. It can be seen that when the loading step is doubled, the external

force is not just doubled which implies that the system demonstrates a nonlinear

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Figure 6.7: Simulation of soft-tissue deformation using nonlinear analysis: (A)Original mesh. (B) Deformed mesh after 200 time steps. (C) Deformed mesh after400 time steps. (D) Comparison between A and C.

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Figure 6.8: Simulation of soft-tissue relaxation from a deformed configuration us-ing nonlinear analysis: (A) Deformed mesh. (B) Mesh configuration 50 time stepsafter removing the load. (C) Mesh relaxed to the original configuration.

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6.3 Steps and Results of 2D Soft-tissue Cutting Simu-

lations

The nonlinear analysis demonstrated in the previous section is employed to sim-

ulate cutting in soft-tissues. In order to let the simulated object deform prior to

cutting and allow progressive cutting, the algorithm shown in Fig. 6.11 is devel-

oped. It can be seen that the types of the interactions between the tool tip and the

simulated soft-tissue model are divided into three cases. The first case is free mo-

tion when there is no contact between the tool tip and the two-dimensional model.

When there is contact between the tool tip and the object, the simulation is divided

into two cases including deformation and cutting. A detailed explanation of the

main blocks is given next.

6.3.1 Main Blocks in the Soft-tissue Cutting Algorithm

• Off-line computations and initialization: This step involves generating the

mesh, getting boundary condition nodes from the user, calculating strain-

displacement matrix, diagonal mass matrix and area for each element, as-

sembling the global mass matrix, and initializing the displacement field. The

details of these calculations have been explained earlier in the chapter.

• Is there a collision?: Since a simple two-dimensional circle is used to model

the object, detecting a collision between the tool tip and the object is done

by determining if the point that represents the tool tip is inside the radius

of the circle or not. Obviously, more sophisticated collision detection routine

should be employed for interaction with object of arbitrary shape.

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Off-line computationsand initialization

Get initial tooltip position

Initialize case tofree motion

Checkcase

Isthere acollision

?

Free Motion Contact

Isthere acollision

?

Evaluate currentnodes position

Meshrelaxation

Change caseto contact

First contactremeshing and

mesh relaxation

YesNoNo Yes

Change caseto free motion

Meshrelaxation

Evaluate elastic andexternal forces

Evaluate currentnodes position

Performcutting

Increment tooltip position

Isexternalforce >thres.

?

YesNo

Figure 6.11: Main blocks of the cutting algorithm

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• First contact remeshing and mesh relaxation: When the first contact be-

tween the tool tip and the object is detected, local remeshing should be per-

formed by moving the closest boundary node to the contact point and ap-

plying the load on this node (load node). The strain-displacement matrix

and area of the elements that contain the load node should be recalculated.

After remeshing, few steps are performed in order to locally relax the mesh

by calculating the deformation field of the nodes in the elements that con-

tain the load node when the tool is just in contact with the object. A higher

damping is used during these steps in order to minimize the oscillations and

get to an equilibrium state faster. In order to prepare for the following time

steps, the deformation field and current nodes position of the whole object

are calculated when the load node is placed at the current tool tip position.

• Mesh relaxation: This step involves the calculations to evaluate the displace-

ment field when there is no external load applied to the object. These calcula-

tions include the evaluation of the elastic force vector for nonlinear analysis

as explained in Sec. 6.2.1 followed by the evaluation of the displacement field

using Eq. 3.27. The position of each node at the current time step can be cal-

culated by adding the obtained displacement field to the original position of

each node.

• Evaluate current nodes position: To compute the position of the nodes in

the mesh at the current time step, the same calculations performed for the

mesh relaxation above should be repeated with the addition of computing

the external force vector on the load node using Eq. 6.15. Therefore, this step

computes the deformation in the object caused by the applied external load.

105


• Is external force > threshold?: To determine if a cutting procedure should

be performed or not, the evaluated external force on the load node is decom-

posed into two components. The component in the tool direction is com-

pared with a threshold value that represents the toughness of the soft-tissue.

If the external force is less than the threshold value, the object is deformed

and the deformation field is computed to update the nodes positions. On the

other hand, if the external force exceeds the predefined threshold value, the

cutting procedure is initiated.

• Perform cutting: The algorithm that was proposed in Chapter 5 is used to

perform cutting.

6.3.2 Simulation Results of Soft-tissue Cutting

In order to illustrate the performance of the cutting algorithm explained above, a

few scenarios will be shown bellow for different cases. In all these cases a constant

threshold value of 40 N is assumed to be the maximum external force that can be

applied to the soft-tissue before a cut occurs. As in the cases of linear and nonlinear

analysis, a time step of 0.001 and α of 0.05 are utilized in the simulations. The

initial position of the tool tip should be specified before the start of the simulation

by specifying the x and y coordinates of the point which should be outside the

mesh. The speed and direction of the load is also specified by choosing the loading

(tool tip displacement) in the x and y directions for each time step. Using trial and

error, it was found out that element quality of 0.2 guaranties numerical stability

for a time step of 0.001. Therefore, the local remeshing algorithm must ensure that

the quality of the elements are higher than 0.2. It should be noted that the tool

106


trajectory is assumed to be constant throughout the simulation which generates a

straight line cut. By simple modifications to the algorithm employed, a curved cut

can be easily generated.

Case 1: Cutting Simulation without Local Remeshing

Fig. 6.12 shows a sequence of steps when tool tip position starts at [0, 0.11] and

is displaced by -0.0001 in the y axis direction at each time step. The sequence

starts with the original undeformed mesh when the first contact between the object

and the tool tip is detected. The second frame shows the node snapping of the

closest boundary node (Node 53) to the contact point and assuming Node 53 to

be the loading node. The third frame of Fig. 6.12 illustrates the deformed mesh

before cutting occurs because the value of the external force on Node 53 in the tool

direction is smaller than the chosen threshold value of 40N. When the value of the

external force on Node 53 reaches the threshold value, the cutting procedure is

performed by snapping the next internal node that makes the smallest angle with

the cut path (Node 52) to the cut path and creating an a new node (Node 89) at the

loading node (Node 51) in order to separate the elements as shown in the Part D of

the figure. In the following time steps, the snapped node (Node 52) becomes the

new loading node as shown in Part E until the threshold value is reached again.

At that point the next internal node is snapped to the cut path as illustrated in Part

F and a new node generated at Node 52.

The above process is repeated every time the external force crosses the thresh-

old value. Fig. 6.13 shows the configuration of the mesh after 350 time step when

few successive cut have been generated and the zoomed version of the mesh in

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First contact point Closest node First contact remeshing

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New loading node Closest node tocut path

Snapped node

Figure 6.12: Case 1 simulation of soft-tissue cutting using nonlinear analysis: (A)Original mesh when first contact detected. (B) First contact remeshing. (C) De-formed mesh before the first cut. (D) Node snapping and elements separation. (E)New loading before second cut. (F) Node snapping and elements separation.

108


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Figure 6.13: Case1 simulation of soft-tissue cutting after few successive cuts: (A)Mesh after 350 time steps. (B) A zoomed figure of part A to show separated ele-ments on the cut path.

Part A illustrates the separated elements at the cut path with no overlapping be-

tween the elements at the two sides of the tool. It should be noted that in this

case the local remeshing algorithm that is used to prevent the generation of small

elements did not change the topology of the local elements or the position of the

snapped node when cutting procedure was conducted. That is because all the gen-

erated elements after cut node snapping had good quality that did not require the

remeshing procedure to be applied.

Fig. 6.14 shows the profile of the external force on the loading node. The com-

ponent of the force vector along the tool direction that is used as the cutting crite-

rion is shown in the figure. It can be seen that the force starts increasing after the

first contact at time step 101. The external force keeps increasing until it passes the

threshold value of 40 where a cutting procedure occurs and then the force drops

in the following time step. After this drop, the force starts increasing again as

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Exte

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Deformation

First Cut

Cutting instants

Deformationafter cutting

Force dropafter cutting

Figure 6.14: Case 1 force profile of the external force on the loading node along thetool direction over 350 time steps

loading continues on the next loading node until it reaches the threshold for the

second time when the second cut occurs. This pattern is repeated as it can be ob-

served very clearly in Fig. 6.14. The results of simulations and particularly the

pattern of the force are consistent with the experimental results reported in the

literature [1, 66] (see Fig 5.8).

Case 2: Cutting Simulation with Local Retriangulation

To demonstrate the effectiveness of the algorithm developed in Sec. 5.2.2 which

remeshes the local elements in the cutting area when badly shaped elements are

generated after node snapping, the current cases and Case 3 are given as examples.

In this case, the algorithm remeshes the local cutting area by only retriangulating

110


the elements using the Delaunay function in order to generate elements with good

quality according to the criterion introduced in Eq. 5.1. The start position of the

tool tip in this case is [0.08 0.07] and it is displaced by -0.0001 in the x axis direction

at each time step.

Fig. 6.15 shows some of the important steps during the simulation. As in the

previous case, the steps starts with the original undeformed mesh when the first

contact between the object and the tool tip is detected. The second stage shows the

node snapping of the closest boundary node (Node 83) to the contact point and

assuming this node to be the loading node. The third part of Fig. 6.15 illustrates

the deformed mesh before cutting occurs where the value of the external force on

Node 83 in the tool direction is smaller than the threshold value. When the value of

the external force on Node 83 reaches the threshold value, the cutting procedure is

performed by snapping the next internal node that makes the smallest angle with

the cut path (Node 73) to the cut path as shown in Part D. It can be seen that this

node snapping generates a degenerated element that can cause numerical instabil-

ity in the subsequent time steps. To avoid this, the proposed remeshing algorithm

retriangulates the local elements and generates elements with better quality. The

results of the retriangulation is shown in Part E of Fig. 6.15. It can also be observed

that a new node (Node 89) has been created at the previous loading node (Node

83) in order to separate the elements, and that Node 73 becomes the loading node

in the subsequent time steps until the next cut is initiated. The last stage in Part F

displays the configuration of the mesh after 350 time steps when a few successive

cuts have been performed.

Fig. 6.16 illustrates the remeshing procedure by showing a zoomed version of

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Figure 6.15: Case 2 simulation of soft-tissue cutting using nonlinear analysis: (A)Original mesh when first contact detected. (B) First contact remeshing. (C) De-formed mesh before the first cut. (D) Node snapping and degenerated elementgeneration. (E) Local remeshing. (F) Successive cuts and mesh configuration after350 time steps.

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the mesh along with the local elements before and after the remeshing procedure.

The profile of the external force in the tool direction on the loading node for this

case is shown in Fig. 6.17.

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Figure 6.16: A zoomed version of parts D and E from Fig. 6.15 with local elementsillustration: (A) Before remeshing. (B) After remeshing.

Case 3: Cutting Simulation with Local Retriangulation and Node Repositioning

In this case, the local remeshig algorithm requires the movement of the snapping

node in order to generate elements with a quality higher than 0.2 because the delu-

anay function can not provide that. The start position of the tool tip is [0.095 0.05]

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Figure 6.17: Case 2 force profile of the external force on the loading node in thetool direction over 350 time steps

and it is displaced by -0.0001 in the y axis direction at each time step.

Fig. 6.18 shows some of the important steps during the simulation. As in the

previous case, the steps starts with the original undeformed mesh when the first

contact between the object and the tool tip is detected. The second frame displays

the node snapping of the closest boundary node (Node 87) to the contact point and

assuming this node to be the loading node. The third part of Fig. 6.18 illustrates

the deformed mesh before cutting occurs. When the value of the external force

on Node 87 reaches the threshold value, the cutting procedure is performed by

snapping the next internal node that makes the smallest angle with the cut path

(Node 80) to the cut path as shown in Part D. It can be seen that this node snap-

ping generates a degenerated element that can cause numerical instability in the

following time steps. Therefore, the remeshing algorithm first retriangulates the

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local elements; however, retriangulation does not generate elements with a quality

higher than 0.2. As a result of this, the snapped node (Node 80) is moved by a

resultant force dependent on the length of the edges connected to the node until

the elements satisfy the quality measure as was explained in Sec 5.2.2.

Fig. 6.19 illustrates the remeshing procedure by showing the local elements at

a sequence of steps until the elements have the required quality. Part E of Fig. 6.15

displays the mesh after the local remeshing and generating a new node (Node 89)

at the previous loading node (Node 87) in order to separate the elements. Node

80 becomes the loading node in the following time steps until the next cut is pre-

formed. The last frame shows the configuration of the mesh after 350 time steps

when a few successive cuts have been performed. The profile of the external force

in the tool direction on the loading node for this case is shown in Fig. 6.20.

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Figure 6.18: Case 3 simulation of soft-tissue cutting using nonlinear analysis: (A)Original mesh when first contact detected. (B) First contact remeshing. (C) De-formed mesh before the first cut. (D) Node snapping and degenerated elementgeneration. (E) Local remeshing. (F) Successive cuts and mesh configuration after350 time steps.

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Figure 6.19: A zoomed version of the local remeshing from part D to E in Fig. 6.18with local elements illustration: (A) Local elements before remeshing. (B,C,D) Theremeshing procedure. (E) Local elements after remeshing.

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Figure 6.20: Case 3 force profile of the external force on the loading node in thetool direction over 350 time steps

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Chapter 7

Conclusions and Future Work

7.1 Conclusions

Virtual reality-based surgical simulators are emerging as a promising alternative

to the conventional means of training. They can also be used for real-time pre-

operative planning of surgical procedures (e.g. percutaneous therapy) as well

as real-time (semi)-autonomous robotic execution of surgical tasks/procedures.

These simulators provide numerous advantages to doctors and patients at the

same time. They offer flexibility during training by allowing surgeons to practice

on virtual patients as they would operate on real patients with realistic sensory

feedback. The ability to adjust the tissue properties and operation scenarios in

these simulators without being concerned about safety issues is critical for the ini-

tial stages of training. Also, they provide a number of other advantages including

the ability to adjust the task difficulty level, quantitatively measure the trainee’s

progress, and provide active guidance. Many medical procedures involve some

119


form of instrument-soft tissue interaction and therefore, models of the human or-

gans soft-tissue that can realistically reproduce haptic (force) and deformation re-

sponse are needed. The complicated mechanical behavior of soft-tissues should

be taken into account when modeling the deformations and interaction forces be-

tween surgical tool and tissues during the operation. There is consensus in the

literature that linear models are insufficient for representing soft-tissue deforma-

tion response and that a nonlinear model should be employed for this purpose.

The contributions of this thesis in using nonlinear Finite Element analysis to

model the mechanical behavior of tissues during tool-tissue interaction in two

common surgical tasks, i.e. palpation and cutting. Geometrical and material non-

linearities are modeled. An incompressible nonlinear material is modeled using a

nonlinear stress-strain measure based on a modified version of the Ogden hypere-

lastic constitutive equation. A penalty term has been added to the energy function

to enforce the incomprehensibility of the tissue material. The total Lagrangian for-

mulation with an explicit dynamic analysis is employed in the implementation.

An algorithm for the modeling of soft-tissue cutting procedures utilizing the

nonlinear FE analysis with the Ogden-based constitutive equation has also been

proposed. Element separation and node snapping is used to introduce the cut into

the mesh. Node snapping can create degenerated elements with small size which

can cause numerical instability in the simulation because of violating the time step

criterion for explicit time integration and generating infinite deformations by zero

area force calculation. Therefore, when such elements are generated after node

snapping, they have to be treated in order to maintain the numerical stability of

the simulation. In this thesis, this is achieved by remeshing the local elements

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when degenerated elements are created. The remeshing process includes retri-

angulation of the elements using Delaunay function and/or moving the snapped

node as needed in order to generate elements with the required quality.

7.2 Future Work

To enhance the simulation process and be able to develop a haptic-enabled surgery

simulator that can incorporate the simulation of surgical prodding and cutting,

some possibilities for future work on this project can include the following:

• Extending the simulation and modeling process to three-dimensional space

which is closer to reality than the two-dimensional analysis implemented in

this thesis. The general cutting algorithm developed can be applied to three-

dimensional objects with simple modifications in the implementation of the

algorithm, e.g. considering surfaces instead of lines.

• Enabling real-time simulation by having real-time deformation and force feed-

back rendering; haptics requires simulation rates in the range of a few hun-

dred Hz whereas the graphics must be updated 30-60 Hz. The nonlinear anal-

ysis combined with a large number of nodes required for accurate modeling

(several thousands nodes may be involved) can be computationally expen-

sive. Meeting the computational requirement of such real-time simulations

is beyond the power of existing conventional CPUs. Regardless of any im-

provement in the calculation speed, real-time calculation of deformations for

meshes of such sizes requires significant computational power. The use of

parallel processing, e.g. based on Field-Programmable Gate Array (FPGA)

121


or Graphics Processing Unit (GPU), can substantially improve the computa-

tion speed and hence enable real-time execution of nonlinear FE deformation

models. The explicit dynamics algorithm that is used in this thesis has the ad-

vantage of element level computations which lend themselves rather neatly

to parallelization.

• Incorporating time-dependent viscoelastic properties observed in the defor-

mation of soft-tissue organs such as the brain into the modeling process [13,

23,33,35,77]. Such soft-tissues have elastic and viscus properties which make

the strain rate dependent on time. Also, hysteresis is observed in the stress-

strain curve of these viscoelastic soft-tissues. The Ogden-based viscoelastic

energy function presented in [13, 35] can be employed for modeling such tis-

sues.

• Performing actual deformation experiments on soft-tissues to obtain the model

parameters for different organs in order to have a simulated tissue response

that matches that of the actual organ.

122

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